Class 11_Chapter 3_Lecture_3p.pdf

“The author makes no claim to the content's originality”

Class XI
Mathematics
Chapter- 3
Trigonometric Functions
Lecture – 3 (Problems)
Dr. Pranav Sharma
Maths Learning Centre. Jalandhar. Maths Learning Centre, Jalandhar Maths Learning Centre, Jalandhar Maths Learning Centre, Jalandhar
Maths Learning Centre, Jalandhar Maths Learning Centre, Jalandhar Maths Learning Centre, Jalandhar
Maths Learning Centre, Jalandhar Maths Learning Centre, Jalandhar Maths Learning Centre, Jalandhar
Maths Learning Centre, Jalandhar Maths Learning Centre, Jalandhar Maths Learning Centre, Jalandhar
Maths Learning Centre, Jalandhar Maths Learning Centre, Jalandhar Maths Learning Centre, Jalandhar
Maths Learning Centre, Jalandhar Maths Learning Centre, Jalandhar Maths Learning Centre, Jalandhar
Maths Learning Centre, Jalandhar Maths Learning Centre, Jalandhar Maths Learning Centre, Jalandhar

youtube.com/@MathematicsOnlineLectures
Find the value of
(i) ??????&#3627408424;&#3627408428; &#3627409362;&#3627409366;&#3627409358;
∘
= ??????&#3627408424;&#3627408428; (&#3627409361;&#3627409364;&#3627409358;
∘
+&#3627409359;&#3627409360;&#3627409358;
∘
)= ??????&#3627408424;&#3627408428; &#3627409359;&#3627409360;&#3627409358;
∘
= ??????&#3627408424;&#3627408428; (&#3627409359;&#3627409366;&#3627409358;
∘
−&#3627409364;&#3627409358;
∘
)=− ??????&#3627408424;&#3627408428; &#3627409364;&#3627409358;
∘
=
−
&#3627409359;
&#3627409360;
.

(ii) &#3627408428;??????&#3627408423; &#3627409359;&#3627409360;&#3627409361;&#3627409358;
∘
= &#3627408428;??????&#3627408423; (&#3627409361;×&#3627409361;&#3627409364;&#3627409358;
∘
+&#3627409359;&#3627409363;&#3627409358;
∘
)= &#3627408428;??????&#3627408423; &#3627409359;&#3627409363;&#3627409358;
∘
= &#3627408428;??????&#3627408423; (&#3627409359;&#3627409366;&#3627409358;
∘
−&#3627409361;&#3627409358;
∘
)=
&#3627408428;??????&#3627408423; &#3627409361;&#3627409358;
∘
=
&#3627409359;
&#3627409360;
.

(iii) ??????&#3627408424;&#3627408429; (−&#3627409359;&#3627409361;&#3627409363;
∘
)=− ??????&#3627408424;&#3627408429; &#3627409359;&#3627409361;&#3627409363;
∘
=− ??????&#3627408424;&#3627408429; (&#3627409367;&#3627409358;
∘
+&#3627409362;&#3627409363;
∘
)= &#3627408429;??????&#3627408423; &#3627409362;&#3627409363;
∘
=&#3627409359;.

Maths Learning Centre, Jalandhar Maths Learning Centre, Jalandhar Maths Learning Centre, Jalandhar
Maths Learning Centre, Jalandhar Maths Learning Centre, Jalandhar Maths Learning Centre, Jalandhar
Maths Learning Centre, Jalandhar Maths Learning Centre, Jalandhar Maths Learning Centre, Jalandhar
Maths Learning Centre, Jalandhar Maths Learning Centre, Jalandhar Maths Learning Centre, Jalandhar
Maths Learning Centre, Jalandhar Maths Learning Centre, Jalandhar Maths Learning Centre, Jalandhar
Maths Learning Centre, Jalandhar Maths Learning Centre, Jalandhar Maths Learning Centre, Jalandhar
Maths Learning Centre, Jalandhar Maths Learning Centre, Jalandhar Maths Learning Centre, Jalandhar

youtube.com/@MathematicsOnlineLectures
(iv) ??????&#3627408424;&#3627408428;???????????? (−&#3627409359;&#3627409362;&#3627409359;&#3627409358;
∘
)=− ??????&#3627408424;&#3627408428;???????????? &#3627409359;&#3627409362;&#3627409359;&#3627409358;
∘
=− ??????&#3627408424;&#3627408428;???????????? (&#3627409362;×&#3627409361;&#3627409364;&#3627409358;
∘
−&#3627409361;&#3627409358;
∘
)
= ??????&#3627408424;&#3627408428;???????????? &#3627409361;&#3627409358;
∘
=&#3627409360;.

(v) ??????&#3627408424;&#3627408428; (−&#3627409366;&#3627409365;&#3627409358;
∘
)= ??????&#3627408424;&#3627408428; &#3627409366;&#3627409365;&#3627409358;
∘
= ??????&#3627408424;&#3627408428; (&#3627409360;×&#3627409361;&#3627409364;&#3627409358;
∘
+&#3627409359;&#3627409363;&#3627409358;
∘
)= ??????&#3627408424;&#3627408428; &#3627409359;&#3627409363;&#3627409358;
∘

= ??????&#3627408424;&#3627408428; (&#3627409359;&#3627409366;&#3627409358;
∘
−&#3627409361;&#3627409358;
∘
)=− ??????&#3627408424;&#3627408428; &#3627409361;&#3627409358;
∘
=
−√&#3627409361;
&#3627409360;

(vi) &#3627408429;??????&#3627408423; &#3627409361;&#3627409361;&#3627409358;
∘
= &#3627408429;??????&#3627408423; (&#3627409361;&#3627409364;&#3627409358;
∘
−&#3627409361;&#3627409358;
∘
)=− &#3627408429;??????&#3627408423; &#3627409361;&#3627409358;
∘
=−
&#3627409359;
√&#3627409361;
.

Maths Learning Centre, Jalandhar Maths Learning Centre, Jalandhar Maths Learning Centre, Jalandhar
Maths Learning Centre, Jalandhar Maths Learning Centre, Jalandhar Maths Learning Centre, Jalandhar
Maths Learning Centre, Jalandhar Maths Learning Centre, Jalandhar Maths Learning Centre, Jalandhar
Maths Learning Centre, Jalandhar Maths Learning Centre, Jalandhar Maths Learning Centre, Jalandhar
Maths Learning Centre, Jalandhar Maths Learning Centre, Jalandhar Maths Learning Centre, Jalandhar
Maths Learning Centre, Jalandhar Maths Learning Centre, Jalandhar Maths Learning Centre, Jalandhar
Maths Learning Centre, Jalandhar Maths Learning Centre, Jalandhar Maths Learning Centre, Jalandhar

youtube.com/@MathematicsOnlineLectures
Find the values of all trigonometric functions of (i) &#3627409359;&#3627409360;&#3627409358;
∘
(ii) &#3627409359;&#3627409363;&#3627409358;
∘
.
(i) &#3627408428;??????&#3627408423; &#3627409359;&#3627409360;&#3627409358;
∘
= &#3627408428;??????&#3627408423; (&#3627409359;&#3627409366;&#3627409358;
∘
−&#3627409364;&#3627409358;
∘
)= &#3627408428;??????&#3627408423; &#3627409364;&#3627409358;
∘
=
√&#3627409361;
&#3627409360;
;
??????&#3627408424;&#3627408428; &#3627409359;&#3627409360;&#3627409358;
∘
= ??????&#3627408424;&#3627408428; (&#3627409359;&#3627409366;&#3627409358;
∘
−&#3627409364;&#3627409358;
∘
)=− ??????&#3627408424;&#3627408428; &#3627409364;&#3627409358;
∘
=
−&#3627409359;
&#3627409360;
;
&#3627408429;??????&#3627408423; &#3627409359;&#3627409360;&#3627409358;
∘
=
&#3627408428;??????&#3627408423; &#3627409359;&#3627409360;&#3627409358;
∘
??????&#3627408424;&#3627408428; &#3627409359;&#3627409360;&#3627409358;
∘
={
√&#3627409361;
&#3627409360;
× (−&#3627409360;)}=−√&#3627409361;; ??????&#3627408424;&#3627408428;???????????? &#3627409359;&#3627409360;&#3627409358;
∘
=
&#3627409359;
&#3627408428;??????&#3627408423; &#3627409359;&#3627409360;&#3627409358;
∘
=
&#3627409360;
√&#3627409361;
;
&#3627408428;???????????? &#3627409359;&#3627409360;&#3627409358;
∘
=
&#3627409359;
??????&#3627408424;&#3627408428; &#3627409359;&#3627409360;&#3627409358;
∘
=−&#3627409360;; ??????&#3627408424;&#3627408429; &#3627409359;&#3627409360;&#3627409358;
∘
=
&#3627409359;
&#3627408429;??????&#3627408423; &#3627409359;&#3627409360;&#3627409358;
∘
=
−&#3627409359;
√&#3627409361;
.

(ii) &#3627408428;??????&#3627408423; &#3627409359;&#3627409363;&#3627409358;
∘
= &#3627408428;??????&#3627408423; (&#3627409359;&#3627409366;&#3627409358;
∘
−&#3627409361;&#3627409358;
∘
)= &#3627408428;??????&#3627408423; &#3627409361;&#3627409358;
∘
=
&#3627409359;
&#3627409360;
;
??????&#3627408424;&#3627408428; &#3627409359;&#3627409363;&#3627409358;
∘
= ??????&#3627408424;&#3627408428; (&#3627409359;&#3627409366;&#3627409358;
∘
−&#3627409361;&#3627409358;
∘
)=− ??????&#3627408424;&#3627408428; &#3627409361;&#3627409358;
∘
=
−√&#3627409361;
&#3627409360;
;
&#3627408429;??????&#3627408423; &#3627409359;&#3627409363;&#3627409358;
∘
=
&#3627408428;??????&#3627408423; &#3627409359;&#3627409363;&#3627409358;
∘
??????&#3627408424;&#3627408428; &#3627409359;&#3627409363;&#3627409358;
∘
=(
&#3627409359;
&#3627409360;
×
&#3627409360;
−√&#3627409361;
)=
−&#3627409359;
√&#3627409361;
; ??????&#3627408424;&#3627408429; &#3627409359;&#3627409363;&#3627409358;
∘
=
&#3627409359;
&#3627408429;??????&#3627408423; &#3627409359;&#3627409363;&#3627409358;
∘
=−√&#3627409361;;
&#3627408428;???????????? &#3627409359;&#3627409363;&#3627409358;
∘
=
&#3627409359;
??????&#3627408424;&#3627408428; &#3627409359;&#3627409363;&#3627409358;
∘
=
−&#3627409360;
√&#3627409361;
; ??????&#3627408424;&#3627408428;???????????? &#3627409359;&#3627409363;&#3627409358;
∘
=
&#3627409359;
&#3627408428;??????&#3627408423; &#3627409359;&#3627409363;&#3627409358;
∘
=&#3627409360;.

Maths Learning Centre, Jalandhar Maths Learning Centre, Jalandhar Maths Learning Centre, Jalandhar
Maths Learning Centre, Jalandhar Maths Learning Centre, Jalandhar Maths Learning Centre, Jalandhar
Maths Learning Centre, Jalandhar Maths Learning Centre, Jalandhar Maths Learning Centre, Jalandhar
Maths Learning Centre, Jalandhar Maths Learning Centre, Jalandhar Maths Learning Centre, Jalandhar
Maths Learning Centre, Jalandhar Maths Learning Centre, Jalandhar Maths Learning Centre, Jalandhar
Maths Learning Centre, Jalandhar Maths Learning Centre, Jalandhar Maths Learning Centre, Jalandhar
Maths Learning Centre, Jalandhar Maths Learning Centre, Jalandhar Maths Learning Centre, Jalandhar

youtube.com/@MathematicsOnlineLectures
Show that &#3627408428;??????&#3627408423; &#3627409359;&#3627409358;&#3627409363;
∘
+ ??????&#3627408424;&#3627408428; &#3627408421;&#3627409358;&#3627409363;
∘
=
&#3627409359;
√&#3627409360;
.
????????????&#3627408402;= &#3627408428;??????&#3627408423; &#3627409359;&#3627409358;&#3627409363;
∘
+ ??????&#3627408424;&#3627408428; &#3627409359;&#3627409358;&#3627409363;
∘
= &#3627408428;??????&#3627408423; (&#3627409364;&#3627409358;
∘
+&#3627409362;&#3627409363;
∘
)+ ??????&#3627408424;&#3627408428; (&#3627409364;&#3627409358;
∘
+&#3627409362;&#3627409363;
∘
)
=( &#3627408428;??????&#3627408423; &#3627409364;&#3627409358;
∘
??????&#3627408424;&#3627408428; &#3627409362;&#3627409363;
∘
+ ??????&#3627408424;&#3627408428; &#3627409364;&#3627409358;
∘
&#3627408428;??????&#3627408423;&#3627409362;&#3627409363;
∘
)+( ??????&#3627408424;&#3627408428; &#3627409364;&#3627409358;
∘
??????&#3627408424;&#3627408428; &#3627409362;&#3627409363;
∘
−&#3627408428;??????&#3627408423;&#3627409364;&#3627409358;
∘
&#3627408428;??????&#3627408423; &#3627409362;&#3627409363;
∘
)
={(
√&#3627409361;
&#3627409360;
×
&#3627409359;
√&#3627409360;
)+(
&#3627409359;
&#3627409360;
×
&#3627409359;
√&#3627409360;
)}+{(
&#3627409359;
&#3627409360;
×
&#3627409359;
√&#3627409360;
)−(
√&#3627409361;
&#3627409360;
×
&#3627409359;
√&#3627409360;
)} =(
√&#3627409361;
&#3627409360;√&#3627409360;
+
&#3627409359;
&#3627409360;√&#3627409360;
+
&#3627409359;
&#3627409360;√&#3627409360;
−
√&#3627409361;
&#3627409360;√&#3627409360;
)=
&#3627409359;
√&#3627409360;
=
&#3627408401;??????&#3627408402;.

Maths Learning Centre, Jalandhar Maths Learning Centre, Jalandhar Maths Learning Centre, Jalandhar
Maths Learning Centre, Jalandhar Maths Learning Centre, Jalandhar Maths Learning Centre, Jalandhar
Maths Learning Centre, Jalandhar Maths Learning Centre, Jalandhar Maths Learning Centre, Jalandhar
Maths Learning Centre, Jalandhar Maths Learning Centre, Jalandhar Maths Learning Centre, Jalandhar
Maths Learning Centre, Jalandhar Maths Learning Centre, Jalandhar Maths Learning Centre, Jalandhar
Maths Learning Centre, Jalandhar Maths Learning Centre, Jalandhar Maths Learning Centre, Jalandhar
Maths Learning Centre, Jalandhar Maths Learning Centre, Jalandhar Maths Learning Centre, Jalandhar

youtube.com/@MathematicsOnlineLectures
&#3627408428;??????&#3627408423; &#3627409359;&#3627409363;
∘
= &#3627408428;??????&#3627408423; (&#3627409362;&#3627409363;
∘
−&#3627409361;&#3627409358;
∘
)= &#3627408428;??????&#3627408423; &#3627409362;&#3627409363;
∘
??????&#3627408424;&#3627408428; &#3627409361;&#3627409358;
∘
− ??????&#3627408424;&#3627408428; &#3627409362;&#3627409363;
∘
&#3627408428;??????&#3627408423; &#3627409361;&#3627409358;
∘

=(
&#3627409359;
√&#3627409360;
×
√&#3627409361;
&#3627409360;
)−(
&#3627409359;
√&#3627409360;
×
&#3627409359;
&#3627409360;
)=
(√&#3627409361;−&#3627409359;)
&#3627409360;√&#3627409360;
.

??????&#3627408424;&#3627408428; &#3627409359;&#3627409363;
∘
= ??????&#3627408424;&#3627408428; (&#3627409362;&#3627409363;
∘
−&#3627409361;&#3627409358;
∘
)= ??????&#3627408424;&#3627408428; &#3627409362;&#3627409363;
∘
??????&#3627408424;&#3627408428; &#3627409361;&#3627409358;
∘
+ &#3627408428;??????&#3627408423; &#3627409362;&#3627409363;
∘
&#3627408428;??????&#3627408423; &#3627409361;&#3627409358;
∘

=(
&#3627409359;
√&#3627409360;
×
√&#3627409361;
&#3627409360;
)+(
&#3627409359;
√&#3627409360;
×
&#3627409359;
&#3627409360;
)=
(√&#3627409361;+&#3627409359;)
&#3627409360;√&#3627409360;
.

&#3627408429;??????&#3627408423; &#3627409359;&#3627409363;
∘
= &#3627408429;??????&#3627408423; (&#3627409362;&#3627409363;
∘
−&#3627409361;&#3627409358;
∘
)=
&#3627408429;??????&#3627408423; &#3627409362;&#3627409363;
∘
− &#3627408429;??????&#3627408423; &#3627409361;&#3627409358;
∘
&#3627409359;+ &#3627408429;??????&#3627408423; &#3627409362;&#3627409363;
∘
&#3627408429;??????&#3627408423; &#3627409361;&#3627409358;
∘
=
(&#3627409359;−
&#3627409359;
√&#3627409361;
)
&#3627409359;+(&#3627409359;×
&#3627409359;
√&#3627409361;
)
=
(√&#3627409361;−&#3627409359;)
(√&#3627409361;+&#3627409359;)
.

Maths Learning Centre, Jalandhar Maths Learning Centre, Jalandhar Maths Learning Centre, Jalandhar
Maths Learning Centre, Jalandhar Maths Learning Centre, Jalandhar Maths Learning Centre, Jalandhar
Maths Learning Centre, Jalandhar Maths Learning Centre, Jalandhar Maths Learning Centre, Jalandhar
Maths Learning Centre, Jalandhar Maths Learning Centre, Jalandhar Maths Learning Centre, Jalandhar
Maths Learning Centre, Jalandhar Maths Learning Centre, Jalandhar Maths Learning Centre, Jalandhar
Maths Learning Centre, Jalandhar Maths Learning Centre, Jalandhar Maths Learning Centre, Jalandhar
Maths Learning Centre, Jalandhar Maths Learning Centre, Jalandhar Maths Learning Centre, Jalandhar

youtube.com/@MathematicsOnlineLectures

&#3627408428;??????&#3627408423; &#3627409365;&#3627409363;
∘
= &#3627408428;??????&#3627408423; (&#3627409362;&#3627409363;
∘
+&#3627409361;&#3627409358;
∘
)= &#3627408428;??????&#3627408423; &#3627409362;&#3627409363;
∘
??????&#3627408424;&#3627408428; &#3627409361;&#3627409358;
∘
+ ??????&#3627408424;&#3627408428; &#3627409362;&#3627409363;
∘
&#3627408428;??????&#3627408423; &#3627409361;&#3627409358;
∘

=(
&#3627409359;
√&#3627409360;
×
√&#3627409361;
&#3627409360;
)+(
&#3627409359;
√&#3627409360;
×
&#3627409359;
&#3627409360;
)=
(√&#3627409361;+&#3627409359;)
&#3627409360;√&#3627409360;
.

??????&#3627408424;&#3627408428; &#3627409365;&#3627409363;
∘
= ??????&#3627408424;&#3627408428; (&#3627409362;&#3627409363;
∘
+&#3627409361;&#3627409358;
∘
)= ??????&#3627408424;&#3627408428; &#3627409362;&#3627409363;
∘
??????&#3627408424;&#3627408428; &#3627409361;&#3627409358;
∘
− &#3627408428;??????&#3627408423; &#3627409362;&#3627409363;
∘
&#3627408428;??????&#3627408423; &#3627409361;&#3627409358;
∘

=(
&#3627409359;
√&#3627409360;
×
√&#3627409361;
&#3627409360;
)−(
&#3627409359;
√&#3627409360;
×
&#3627409359;
&#3627409360;
)=
(√&#3627409361;−&#3627409359;)
&#3627409360;√&#3627409360;
.

&#3627408429;??????&#3627408423; &#3627409365;&#3627409363;
∘
= &#3627408429;??????&#3627408423; (&#3627409362;&#3627409363;
∘
+&#3627409361;&#3627409358;
∘
) =
&#3627408429;??????&#3627408423; &#3627409362;&#3627409363;
∘
+ &#3627408429;??????&#3627408423; &#3627409361;&#3627409358;
∘
&#3627409359;− &#3627408429;??????&#3627408423; &#3627409362;&#3627409363;
∘
&#3627408429;??????&#3627408423; &#3627409361;&#3627409358;
=
(&#3627409359;+
&#3627409359;
√&#3627409361;
)
(&#3627409359;−
&#3627409359;
√&#3627409361;
)
=
(√&#3627409361;+&#3627409359;)
(√&#3627409361;−&#3627409359;)
.

&#3627408429;??????&#3627408423;
&#3627409359;&#3627409361;??????
&#3627409359;&#3627409360;
= &#3627408429;??????&#3627408423; (??????+
??????
&#3627409359;&#3627409360;
)= &#3627408429;??????&#3627408423;
??????
&#3627409359;&#3627409360;
= &#3627408429;??????&#3627408423; (
??????
&#3627409361;
−
??????
&#3627409362;
)=
&#3627408429;??????&#3627408423;
??????
&#3627409361;
− &#3627408429;??????&#3627408423;
??????
&#3627409362;
&#3627409359;+ &#3627408429;??????&#3627408423;
??????
&#3627409361;
&#3627408429;??????&#3627408423;
??????
&#3627409362;
=
√&#3627409361;−&#3627409359;
&#3627409359;+√&#3627409361;

=
(√&#3627409361;−&#3627409359;)
(√&#3627409361;+&#3627409359;)
×
(√&#3627409361;−&#3627409359;)
(√&#3627409361;−&#3627409359;)
=
(√&#3627409361;−&#3627409359;)
&#3627409360;
(&#3627409361;−&#3627409359;)
=
(&#3627409361;+&#3627409359;−&#3627409360;√&#3627409361;)
&#3627409360;
=
(&#3627409362;−&#3627409360;√&#3627409361;)
&#3627409360;
=(&#3627409360;−√&#3627409361;) . Maths Learning Centre, Jalandhar Maths Learning Centre, Jalandhar Maths Learning Centre, Jalandhar
Maths Learning Centre, Jalandhar Maths Learning Centre, Jalandhar Maths Learning Centre, Jalandhar
Maths Learning Centre, Jalandhar Maths Learning Centre, Jalandhar Maths Learning Centre, Jalandhar
Maths Learning Centre, Jalandhar Maths Learning Centre, Jalandhar Maths Learning Centre, Jalandhar
Maths Learning Centre, Jalandhar Maths Learning Centre, Jalandhar Maths Learning Centre, Jalandhar
Maths Learning Centre, Jalandhar Maths Learning Centre, Jalandhar Maths Learning Centre, Jalandhar
Maths Learning Centre, Jalandhar Maths Learning Centre, Jalandhar Maths Learning Centre, Jalandhar

youtube.com/@MathematicsOnlineLectures
Prove that
(i) &#3627408428;??????&#3627408423; &#3627409365;&#3627409358;
∘
??????&#3627408424;&#3627408428; &#3627409359;&#3627409358;
∘
− ??????&#3627408424;&#3627408428; &#3627409365;&#3627409358;
∘
&#3627408428;??????&#3627408423; &#3627409359;&#3627409358;
∘
=
√&#3627409361;
&#3627409360;

(ii) ??????&#3627408424;&#3627408428; &#3627409363;&#3627409358;
∘
??????&#3627408424;&#3627408428; &#3627409359;&#3627409358;
∘
− &#3627408428;??????&#3627408423; &#3627409363;&#3627409358;
∘
&#3627408428;??????&#3627408423; &#3627409359;&#3627409358;
∘
=
&#3627409359;
&#3627409360;

(iii) ??????&#3627408424;&#3627408428; &#3627409366;&#3627409358;
∘
??????&#3627408424;&#3627408428; &#3627409360;&#3627409358;
∘
+ &#3627408428;??????&#3627408423; &#3627409366;&#3627409358;
∘
&#3627408428;??????&#3627408423; &#3627409360;&#3627409358;
∘
=
&#3627409359;
&#3627409360;

(iv) &#3627408428;??????&#3627408423; &#3627409361;&#3627409364;
∘
??????&#3627408424;&#3627408428; &#3627409367;
∘
+ ??????&#3627408424;&#3627408428; &#3627409361;&#3627409364;
∘
&#3627408428;??????&#3627408423; &#3627409367;
∘
=
&#3627409359;
√&#3627409360;

(i) We know that &#3627408428;??????&#3627408423; ?????? ??????&#3627408424;&#3627408428; ??????− ??????&#3627408424;&#3627408428; ?????? &#3627408428;??????&#3627408423; ??????= &#3627408428;??????&#3627408423; (??????−??????) .
Putting ?????? =&#3627409365;&#3627409358;
∘
and ??????=&#3627409359;&#3627409358;
∘
in the this identity, we get
&#3627408428;??????&#3627408423; &#3627409365;&#3627409358;
∘
??????&#3627408424;&#3627408428; &#3627409359;&#3627409358;
∘
− ??????&#3627408424;&#3627408428; &#3627409365;&#3627409358;
∘
&#3627408428;??????&#3627408423; &#3627409359;&#3627409358;
∘
= &#3627408428;??????&#3627408423; (&#3627409365;&#3627409358;
∘
−&#3627409359;&#3627409358;
∘
)= &#3627408428;??????&#3627408423; &#3627409364;&#3627409358;
∘
=
√&#3627409361;
&#3627409360;
.

(ii) We know that ??????&#3627408424;&#3627408428; ?????? ??????&#3627408424;&#3627408428; ??????− &#3627408428;??????&#3627408423; ?????? &#3627408428;??????&#3627408423; ??????= ??????&#3627408424;&#3627408428; (??????+??????) .
Putting6 =&#3627409363;&#3627409358;
∘
and ??????=&#3627409359;&#3627409358;
∘
in the this identity, we get
??????&#3627408424;&#3627408428; &#3627409363;&#3627409358;
∘
??????&#3627408424;&#3627408428; &#3627409359;&#3627409358;
∘
− &#3627408428;??????&#3627408423; &#3627409363;&#3627409358;
∘
&#3627408428;??????&#3627408423; &#3627409359;&#3627409358;
∘
= ??????&#3627408424;&#3627408428; (&#3627409363;&#3627409358;
∘
+&#3627409359;&#3627409358;
∘
)= ??????&#3627408424;&#3627408428; &#3627409364;&#3627409358;
∘
=
&#3627409359;
&#3627409360;
.
Maths Learning Centre, Jalandhar Maths Learning Centre, Jalandhar Maths Learning Centre, Jalandhar
Maths Learning Centre, Jalandhar Maths Learning Centre, Jalandhar Maths Learning Centre, Jalandhar
Maths Learning Centre, Jalandhar Maths Learning Centre, Jalandhar Maths Learning Centre, Jalandhar
Maths Learning Centre, Jalandhar Maths Learning Centre, Jalandhar Maths Learning Centre, Jalandhar
Maths Learning Centre, Jalandhar Maths Learning Centre, Jalandhar Maths Learning Centre, Jalandhar
Maths Learning Centre, Jalandhar Maths Learning Centre, Jalandhar Maths Learning Centre, Jalandhar
Maths Learning Centre, Jalandhar Maths Learning Centre, Jalandhar Maths Learning Centre, Jalandhar

youtube.com/@MathematicsOnlineLectures
(iii) We know that ??????&#3627408424;&#3627408428; ?????? ??????&#3627408424;&#3627408428; ??????+ &#3627408428;??????&#3627408423; ?????? &#3627408428;??????&#3627408423; ??????= ??????&#3627408424;&#3627408428; (??????−??????) .
Putting ?????? =&#3627409366;&#3627409358;
∘
and ??????=&#3627409360;&#3627409358;
∘
in this identity, we get
??????&#3627408424;&#3627408428; &#3627409366;&#3627409358;
∘
??????&#3627408424;&#3627408428; &#3627409360;&#3627409358;
∘
+ &#3627408428;??????&#3627408423; &#3627409366;&#3627409358;
∘
&#3627408428;??????&#3627408423; &#3627409360;&#3627409358;
∘
= ??????&#3627408424;&#3627408428; (&#3627409366;&#3627409358;
∘
−&#3627409360;&#3627409358;
∘
)= ??????&#3627408424;&#3627408428; &#3627409364;&#3627409358;
∘
=
&#3627409359;
&#3627409360;
.

(iv) We know that &#3627408428;??????&#3627408423; ?????? ??????&#3627408424;&#3627408428; ??????+ ??????&#3627408424;&#3627408428; ?????? &#3627408428;??????&#3627408423; ??????= &#3627408428;??????&#3627408423; (??????+??????) .
Putting ?????? =&#3627409361;&#3627409364;
∘
and ??????=&#3627409367;
∘
in this identity, we get
&#3627408428;??????&#3627408423; &#3627409361;&#3627409364;
∘
??????&#3627408424;&#3627408428; &#3627409367;
∘
+ ??????&#3627408424;&#3627408428; &#3627409361;&#3627409364;
∘
&#3627408428;??????&#3627408423; &#3627409367;
∘
=&#3627408428;i&#3627408423;(&#3627409361;&#3627409364;
∘
+&#3627409367;
∘
)= &#3627408428;??????&#3627408423; &#3627409362;&#3627409363;
∘
=
&#3627409359;
√&#3627409360;
.

Maths Learning Centre, Jalandhar Maths Learning Centre, Jalandhar Maths Learning Centre, Jalandhar
Maths Learning Centre, Jalandhar Maths Learning Centre, Jalandhar Maths Learning Centre, Jalandhar
Maths Learning Centre, Jalandhar Maths Learning Centre, Jalandhar Maths Learning Centre, Jalandhar
Maths Learning Centre, Jalandhar Maths Learning Centre, Jalandhar Maths Learning Centre, Jalandhar
Maths Learning Centre, Jalandhar Maths Learning Centre, Jalandhar Maths Learning Centre, Jalandhar
Maths Learning Centre, Jalandhar Maths Learning Centre, Jalandhar Maths Learning Centre, Jalandhar
Maths Learning Centre, Jalandhar Maths Learning Centre, Jalandhar Maths Learning Centre, Jalandhar

youtube.com/@MathematicsOnlineLectures
Prove that &#3627408428;??????&#3627408423; (&#3627409362;&#3627409358;
∘
+??????) ??????&#3627408424;&#3627408428; (&#3627409359;&#3627409358;
∘
+??????)− ??????&#3627408424;&#3627408428; (&#3627409362;&#3627409358;
∘
+??????)&#3627408428;??????&#3627408423;(&#3627409359;&#3627409358;
∘
+??????)=
&#3627409359;
&#3627409360;
.
We know that &#3627408428;??????&#3627408423; &#3627408537; ??????&#3627408424;&#3627408428; &#3627408538;− ??????&#3627408424;&#3627408428; &#3627408537; &#3627408428;??????&#3627408423; &#3627408538;= &#3627408428;??????&#3627408423; (&#3627408537;−&#3627408538;) .
Putting &#3627408537;=(&#3627409362;&#3627409358;
∘
+??????) and &#3627408538;=(&#3627409359;&#3627409358;
∘
+??????) in this identity, we get
&#3627408428;??????&#3627408423; (&#3627409362;&#3627409358;
∘
+??????) ??????&#3627408424;&#3627408428; (&#3627409359;&#3627409358;
∘
+??????)− ??????&#3627408424;&#3627408428; (&#3627409362;&#3627409358;
∘
+??????) &#3627408428;??????&#3627408423; (&#3627409359;&#3627409358;
∘
+??????)
= &#3627408428;??????&#3627408423; [(&#3627409362;&#3627409358;
∘
+??????)−(&#3627409359;&#3627409358;
∘
+??????)]= &#3627408428;??????&#3627408423; &#3627409361;&#3627409358;
∘
=
&#3627409359;
&#3627409360;
.

Prove that ??????&#3627408424;&#3627408428; ??????+ &#3627408428;??????&#3627408423; (&#3627409360;&#3627409365;&#3627409358;
∘
+??????)− &#3627408428;??????&#3627408423; (&#3627409360;&#3627409365;&#3627409358;
∘
−??????)+ ??????&#3627408424;&#3627408428; (&#3627409359;&#3627409366;&#3627409358;
∘
+??????)=&#3627409358;.
We have ??????&#3627408424;&#3627408428; ??????+ &#3627408428;??????&#3627408423; (&#3627409360;&#3627409365;&#3627409358;
∘
+??????)− &#3627408428;??????&#3627408423; (&#3627409360;&#3627409365;&#3627409358;
∘
−??????)+ ??????&#3627408424;&#3627408428; (&#3627409359;&#3627409366;&#3627409358;
∘
+??????)
= ??????&#3627408424;&#3627408428; ??????+ &#3627408428;??????&#3627408423; {&#3627409359;&#3627409366;&#3627409358;
∘
+(&#3627409367;&#3627409358;
∘
+??????)}− &#3627408428;??????&#3627408423; {&#3627409359;&#3627409366;&#3627409358;
∘
+(&#3627409367;&#3627409358;
∘
−??????)}+ ??????&#3627408424;&#3627408428; (&#3627409359;&#3627409366;&#3627409358;
∘
+??????)
= ??????&#3627408424;&#3627408428; ??????− &#3627408428;??????&#3627408423; (&#3627409367;&#3627409358;
∘
+??????)+ &#3627408428;??????&#3627408423; (&#3627409367;&#3627409358;
∘
−??????)− ??????&#3627408424;&#3627408428; ??????
= ??????&#3627408424;&#3627408428; ??????− ??????&#3627408424;&#3627408428; ??????+??????&#3627408424;&#3627408428;??????− ??????&#3627408424;&#3627408428; ??????=&#3627409358;

Maths Learning Centre, Jalandhar Maths Learning Centre, Jalandhar Maths Learning Centre, Jalandhar
Maths Learning Centre, Jalandhar Maths Learning Centre, Jalandhar Maths Learning Centre, Jalandhar
Maths Learning Centre, Jalandhar Maths Learning Centre, Jalandhar Maths Learning Centre, Jalandhar
Maths Learning Centre, Jalandhar Maths Learning Centre, Jalandhar Maths Learning Centre, Jalandhar
Maths Learning Centre, Jalandhar Maths Learning Centre, Jalandhar Maths Learning Centre, Jalandhar
Maths Learning Centre, Jalandhar Maths Learning Centre, Jalandhar Maths Learning Centre, Jalandhar
Maths Learning Centre, Jalandhar Maths Learning Centre, Jalandhar Maths Learning Centre, Jalandhar

youtube.com/@MathematicsOnlineLectures
Prove that
??????&#3627408424;&#3627408428; (&#3627409367;&#3627409358;
∘
+??????) &#3627408428;???????????? (&#3627409360;&#3627409365;&#3627409358;
∘
+??????) &#3627408428;??????&#3627408423; (&#3627409359;&#3627409366;&#3627409358;
∘
+??????)
??????&#3627408424;&#3627408428;???????????? (−??????) ??????&#3627408424;&#3627408428; (&#3627409360;&#3627409365;&#3627409358;
∘
−??????) &#3627408429;??????&#3627408423; (&#3627409359;&#3627409366;&#3627409358;
∘
+??????)
= ??????&#3627408424;&#3627408428; ??????.
????????????&#3627408402;=
??????&#3627408424;&#3627408428; (&#3627409367;&#3627409358;
∘
+??????) &#3627408428;???????????? (&#3627409360;&#3627409365;&#3627409358;
∘
+??????) &#3627408428;??????&#3627408423; (&#3627409359;&#3627409366;&#3627409358;
∘
+??????)
??????&#3627408424;&#3627408428;???????????? (−??????) ??????&#3627408424;&#3627408428; (&#3627409360;&#3627409365;&#3627409358;
∘
−??????) &#3627408429;??????&#3627408423; (&#3627409359;&#3627409366;&#3627409358;
∘
+??????)
=
??????&#3627408424;&#3627408428; (&#3627409367;&#3627409358;
∘
+??????) &#3627408428;???????????? {&#3627409359;&#3627409366;&#3627409358;
∘
+(&#3627409367;&#3627409358;
∘
+??????)} &#3627408428;??????&#3627408423; (&#3627409359;&#3627409366;&#3627409358;
∘
+??????)
− ??????&#3627408424;&#3627408428;???????????? ?????? ??????&#3627408424;&#3627408428; {&#3627409359;&#3627409366;&#3627409358;
∘
+(&#3627409367;&#3627409358;
∘
−??????)} &#3627408429;??????&#3627408423; (&#3627409359;&#3627409366;&#3627409358;
∘
+??????)

=
??????&#3627408424;&#3627408428; (&#3627409367;&#3627409358;
∘
+??????){− &#3627408428;???????????? (&#3627409367;&#3627409358;
∘
+??????)}{− &#3627408428;??????&#3627408423; ??????}
− ??????&#3627408424;&#3627408428;???????????? ??????{− ??????&#3627408424;&#3627408428; (&#3627409367;&#3627409358;
∘
−??????)} &#3627408429;??????&#3627408423; ??????
=
(− &#3627408428;??????&#3627408423; ??????)( ??????&#3627408424;&#3627408428;???????????? ??????)(− &#3627408428;??????&#3627408423; ??????)
(− ??????&#3627408424;&#3627408428;???????????? ??????)(− &#3627408428;??????&#3627408423; ??????) &#3627408429;??????&#3627408423; ??????
= ??????&#3627408424;&#3627408428; ??????=&#3627408401;??????&#3627408402;.

Maths Learning Centre, Jalandhar Maths Learning Centre, Jalandhar Maths Learning Centre, Jalandhar
Maths Learning Centre, Jalandhar Maths Learning Centre, Jalandhar Maths Learning Centre, Jalandhar
Maths Learning Centre, Jalandhar Maths Learning Centre, Jalandhar Maths Learning Centre, Jalandhar
Maths Learning Centre, Jalandhar Maths Learning Centre, Jalandhar Maths Learning Centre, Jalandhar
Maths Learning Centre, Jalandhar Maths Learning Centre, Jalandhar Maths Learning Centre, Jalandhar
Maths Learning Centre, Jalandhar Maths Learning Centre, Jalandhar Maths Learning Centre, Jalandhar
Maths Learning Centre, Jalandhar Maths Learning Centre, Jalandhar Maths Learning Centre, Jalandhar

youtube.com/@MathematicsOnlineLectures
Prove that ??????&#3627408424;&#3627408428; (
&#3627409361;??????
&#3627409360;
+??????) ??????&#3627408424;&#3627408428; (&#3627409360;??????+??????)[ ??????&#3627408424;&#3627408429; (
&#3627409361;??????
&#3627409360;
−??????)+ ??????&#3627408424;&#3627408429; (&#3627409360;??????+??????)]=&#3627409359;.
??????&#3627408424;&#3627408428; (
&#3627409361;??????
&#3627409360;
+??????) ??????&#3627408424;&#3627408428; (&#3627409360;??????+??????)[ ??????&#3627408424;&#3627408429; (
&#3627409361;??????
&#3627409360;
−??????)+ ??????&#3627408424;&#3627408429; (&#3627409360;??????+??????)]
= ??????&#3627408424;&#3627408428; [??????+ (
??????
&#3627409360;
+??????)] ??????&#3627408424;&#3627408428; (&#3627409360;??????+??????)[ ??????&#3627408424;&#3627408429; {??????+ (
??????
&#3627409360;
−??????)}+ ??????&#3627408424;&#3627408429; (&#3627409360;??????+??????)]
=− ??????&#3627408424;&#3627408428; (
??????
&#3627409360;
+??????) ??????&#3627408424;&#3627408428; ??????[ ??????&#3627408424;&#3627408429; (
??????
&#3627409360;
−??????)+ ??????&#3627408424;&#3627408429; ??????]=&#3627408428;??????&#3627408423;?????? ??????&#3627408424;&#3627408428; ??????[ &#3627408429;??????&#3627408423; ??????+ ??????&#3627408424;&#3627408429; ??????]
=( &#3627408428;??????&#3627408423; ?????? ??????&#3627408424;&#3627408428; ??????)[
&#3627408428;??????&#3627408423; ??????
??????&#3627408424;&#3627408428; ??????
+
??????&#3627408424;&#3627408428; ??????
&#3627408428;??????&#3627408423; ??????
]=( &#3627408428;??????&#3627408423; ?????? ??????&#3627408424;&#3627408428; ??????)⋅
(&#3627408428;??????&#3627408423;
&#3627409360;
??????+??????&#3627408424;&#3627408428;
&#3627409360;
??????)
( &#3627408428;??????&#3627408423; ?????? ??????&#3627408424;&#3627408428; ??????)
=&#3627409359;.

Maths Learning Centre, Jalandhar Maths Learning Centre, Jalandhar Maths Learning Centre, Jalandhar
Maths Learning Centre, Jalandhar Maths Learning Centre, Jalandhar Maths Learning Centre, Jalandhar
Maths Learning Centre, Jalandhar Maths Learning Centre, Jalandhar Maths Learning Centre, Jalandhar
Maths Learning Centre, Jalandhar Maths Learning Centre, Jalandhar Maths Learning Centre, Jalandhar
Maths Learning Centre, Jalandhar Maths Learning Centre, Jalandhar Maths Learning Centre, Jalandhar
Maths Learning Centre, Jalandhar Maths Learning Centre, Jalandhar Maths Learning Centre, Jalandhar
Maths Learning Centre, Jalandhar Maths Learning Centre, Jalandhar Maths Learning Centre, Jalandhar

youtube.com/@MathematicsOnlineLectures
Prove that
(??????) &#3627408428;??????&#3627408423;
&#3627409365;??????
&#3627409359;&#3627409360;
??????&#3627408424;&#3627408428;
??????
&#3627409362;
− ??????&#3627408424;&#3627408428;
&#3627409365;??????
&#3627409359;&#3627409360;
&#3627408428;??????&#3627408423;
??????
&#3627409362;
=
√&#3627409361;
&#3627409360;
(????????????) &#3627408428;??????&#3627408423;
??????
&#3627409362;
??????&#3627408424;&#3627408428;
??????
&#3627409359;&#3627409360;
+ ??????&#3627408424;&#3627408428;
??????
&#3627409362;
&#3627408428;??????&#3627408423;
??????
&#3627409359;&#3627409360;
=
√&#3627409361;
&#3627409360;

(iii) ??????&#3627408424;&#3627408428;
&#3627409360;??????
&#3627409361;
??????&#3627408424;&#3627408428;
??????
&#3627409362;
− &#3627408428;??????&#3627408423;
&#3627409360;??????
&#3627409361;
&#3627408428;??????&#3627408423;
??????
&#3627409362;
=
−(√&#3627409361;+&#3627409359;)
&#3627409360;√&#3627409360;

(i) &#3627408428;??????&#3627408423;
&#3627409365;??????
&#3627409359;&#3627409360;
??????&#3627408424;&#3627408428;
??????
&#3627409362;
− ??????&#3627408424;&#3627408428;
&#3627409365;??????
&#3627409359;&#3627409360;
&#3627408428;??????&#3627408423;
??????
&#3627409362;

= &#3627408428;??????&#3627408423; (
&#3627409365;??????
&#3627409359;&#3627409360;
−
??????
&#3627409362;
) [ &#3627408428;??????&#3627408423; &#3627408537; ??????&#3627408424;&#3627408428; &#3627408538;− ??????&#3627408424;&#3627408428; &#3627408537; &#3627408428;??????&#3627408423; &#3627408538;= &#3627408428;??????&#3627408423; (&#3627408537;−&#3627408538;)]=&#3627408428;??????&#3627408423;
??????
&#3627409361;
=
√&#3627409361;
&#3627409360;
.

(ii) &#3627408428;??????&#3627408423;
??????
&#3627409362;
??????&#3627408424;&#3627408428;
??????
&#3627409359;&#3627409360;
+ ??????&#3627408424;&#3627408428;
??????
&#3627409362;
&#3627408428;??????&#3627408423;
??????
&#3627409359;&#3627409360;

= &#3627408428;??????&#3627408423; (
??????
&#3627409362;
+
??????
&#3627409359;&#3627409360;
) [ &#3627408428;??????&#3627408423;&#3627408537; ??????&#3627408424;&#3627408428; &#3627408538;+ ??????&#3627408424;&#3627408428; &#3627408537; &#3627408428;??????&#3627408423; &#3627408538;= &#3627408428;??????&#3627408423; (&#3627408537;+&#3627408538;)]= &#3627408428;??????&#3627408423;
??????
&#3627409361;
=
√&#3627409361;
&#3627409360;
.

(iii) ??????&#3627408424;&#3627408428;
&#3627409360;??????
&#3627409361;
??????&#3627408424;&#3627408428;
??????
&#3627409362;
− &#3627408428;??????&#3627408423;
&#3627409360;??????
&#3627409361;
&#3627408428;??????&#3627408423;
??????
&#3627409362;
= ??????&#3627408424;&#3627408428; (??????−
??????
&#3627409361;
) ??????&#3627408424;&#3627408428;
??????
&#3627409362;
− &#3627408428;??????&#3627408423; (??????−
??????
&#3627409361;
) &#3627408428;??????&#3627408423;
??????
&#3627409362;

=− ??????&#3627408424;&#3627408428;
??????
&#3627409361;
??????&#3627408424;&#3627408428;
??????
&#3627409362;
− &#3627408428;??????&#3627408423;
??????
&#3627409361;
??????&#3627408424;&#3627408428;
??????
&#3627409362;
=(−
&#3627409359;
&#3627409360;
×
&#3627409359;
√&#3627409360;
)−(
√&#3627409361;
&#3627409360;
×
&#3627409359;
√&#3627409360;
)=−{
(√&#3627409361;+&#3627409359;)
&#3627409360;√&#3627409360;
}. Maths Learning Centre, Jalandhar Maths Learning Centre, Jalandhar Maths Learning Centre, Jalandhar
Maths Learning Centre, Jalandhar Maths Learning Centre, Jalandhar Maths Learning Centre, Jalandhar
Maths Learning Centre, Jalandhar Maths Learning Centre, Jalandhar Maths Learning Centre, Jalandhar
Maths Learning Centre, Jalandhar Maths Learning Centre, Jalandhar Maths Learning Centre, Jalandhar
Maths Learning Centre, Jalandhar Maths Learning Centre, Jalandhar Maths Learning Centre, Jalandhar
Maths Learning Centre, Jalandhar Maths Learning Centre, Jalandhar Maths Learning Centre, Jalandhar
Maths Learning Centre, Jalandhar Maths Learning Centre, Jalandhar Maths Learning Centre, Jalandhar

youtube.com/@MathematicsOnlineLectures
Evaluate: (i) &#3627408428;??????&#3627408423;
??????
&#3627409359;&#3627409360;
(ii) &#3627408428;??????&#3627408423;
&#3627409363;??????
&#3627409359;&#3627409360;

(i) &#3627408428;??????&#3627408423;
??????
&#3627409359;&#3627409360;
= &#3627408428;??????&#3627408423; (
??????
&#3627409362;
−
??????
&#3627409364;
)= &#3627408428;??????&#3627408423;
??????
&#3627409362;
??????&#3627408424;&#3627408428;
??????
&#3627409364;
− ??????&#3627408424;&#3627408428;
??????
&#3627409362;
&#3627408428;i&#3627408423;
??????
&#3627409364;

=(
&#3627409359;
√&#3627409360;
×
√&#3627409361;
&#3627409360;
)−(
&#3627409359;
√&#3627409360;
×
&#3627409359;
&#3627409360;
)=
(√&#3627409361;−&#3627409359;)
&#3627409360;√&#3627409360;
.

(ii) &#3627408428;??????&#3627408423;
&#3627409363;??????
&#3627409359;&#3627409360;
= &#3627408428;??????&#3627408423; (
??????
&#3627409362;
+
??????
&#3627409364;
)= &#3627408428;??????&#3627408423;
??????
&#3627409362;
??????&#3627408424;&#3627408428;
??????
&#3627409364;
+ ??????&#3627408424;&#3627408428;
??????
&#3627409362;
&#3627408428;??????&#3627408423;
??????
&#3627409364;

=(
&#3627409359;
√&#3627409360;
×
√&#3627409361;
&#3627409360;
)+(
&#3627409359;
√&#3627409360;
×
&#3627409359;
&#3627409360;
)=
(√&#3627409361;+&#3627409359;)
&#3627409360;√&#3627409360;
.

Maths Learning Centre, Jalandhar Maths Learning Centre, Jalandhar Maths Learning Centre, Jalandhar
Maths Learning Centre, Jalandhar Maths Learning Centre, Jalandhar Maths Learning Centre, Jalandhar
Maths Learning Centre, Jalandhar Maths Learning Centre, Jalandhar Maths Learning Centre, Jalandhar
Maths Learning Centre, Jalandhar Maths Learning Centre, Jalandhar Maths Learning Centre, Jalandhar
Maths Learning Centre, Jalandhar Maths Learning Centre, Jalandhar Maths Learning Centre, Jalandhar
Maths Learning Centre, Jalandhar Maths Learning Centre, Jalandhar Maths Learning Centre, Jalandhar
Maths Learning Centre, Jalandhar Maths Learning Centre, Jalandhar Maths Learning Centre, Jalandhar

youtube.com/@MathematicsOnlineLectures
Prove that
&#3627408428;??????&#3627408423; (&#3627408537;+??????)
&#3627408428;&#3627408422;̇(&#3627408537;−&#3627408538;)
=
&#3627408429;??????&#3627408423; &#3627408537;+ &#3627408429;??????&#3627408423; &#3627408538;
&#3627408429;??????&#3627408423; &#3627408537;− &#3627408429;??????&#3627408423; &#3627408538;
.
We have ????????????&#3627408402;=
&#3627408428;??????&#3627408423; (&#3627408537;+&#3627408538;)
&#3627408428;??????&#3627408423; (&#3627408537;−&#3627408538;)
=
&#3627408428;??????&#3627408527; &#3627408537; ??????&#3627408424;&#3627408428; &#3627408538;+ ??????&#3627408424;&#3627408428; &#3627408537; &#3627408428;??????&#3627408527; &#3627408538;
&#3627408428;??????&#3627408527; &#3627408537; ??????&#3627408424;&#3627408428; &#3627408538;− ??????&#3627408424;&#3627408428; &#3627408537; &#3627408428;??????&#3627408527; &#3627408538;
=
&#3627408429;??????&#3627408423; &#3627408537;+ &#3627408429;??????&#3627408423; &#3627408538;
&#3627408429;??????&#3627408423; &#3627408537;− &#3627408429;??????&#3627408423; &#3627408538;
[dividing &#3627408423;&#3627408430;&#3627408422;. and
denom. by ??????&#3627408424;&#3627408428; &#3627408537; ??????&#3627408424;&#3627408428; &#3627408538;]=&#3627408401;??????&#3627408402;.

Maths Learning Centre, Jalandhar Maths Learning Centre, Jalandhar Maths Learning Centre, Jalandhar
Maths Learning Centre, Jalandhar Maths Learning Centre, Jalandhar Maths Learning Centre, Jalandhar
Maths Learning Centre, Jalandhar Maths Learning Centre, Jalandhar Maths Learning Centre, Jalandhar
Maths Learning Centre, Jalandhar Maths Learning Centre, Jalandhar Maths Learning Centre, Jalandhar
Maths Learning Centre, Jalandhar Maths Learning Centre, Jalandhar Maths Learning Centre, Jalandhar
Maths Learning Centre, Jalandhar Maths Learning Centre, Jalandhar Maths Learning Centre, Jalandhar
Maths Learning Centre, Jalandhar Maths Learning Centre, Jalandhar Maths Learning Centre, Jalandhar

youtube.com/@MathematicsOnlineLectures
If &#3627408428;??????&#3627408423;??????=
&#3627409359;&#3627409363;
&#3627409359;&#3627409365;
and ??????&#3627408424;&#3627408428; ??????=
&#3627409359;&#3627409360;
&#3627409359;&#3627409361;
, where 6 and ?????? both lie in the first quadrant, find the
values of (i) &#3627408428;??????&#3627408423; (??????+??????) , (ii) ??????&#3627408424;&#3627408428; (??????−??????) , (iii) &#3627408429;??????&#3627408423; (??????+??????) .
Given: &#3627408428;??????&#3627408423; ??????=
&#3627409359;&#3627409363;
&#3627409359;&#3627409365;
and ??????&#3627408424;&#3627408428; ??????=
&#3627409359;&#3627409360;
&#3627409359;&#3627409361;

Since ?????? lies in the first quadrant, we have &#3627408428;??????&#3627408423; ??????>&#3627409358;, ??????&#3627408424;&#3627408428; ??????>&#3627409358; and &#3627408429;??????&#3627408423; ??????>&#3627409358;.
Again, since ??????lies in the first quadrant, we have &#3627408428;??????&#3627408423; ??????>&#3627409358;, ??????&#3627408424;&#3627408428; ??????>&#3627409358; and &#3627408429;??????&#3627408423; ??????>
&#3627409358;.
Now, ??????&#3627408528;??????
&#3627409360;
?????? =(&#3627409359;−&#3627408428;??????&#3627408423;
&#3627409360;
??????)=(&#3627409359;−
&#3627409360;&#3627409360;&#3627409363;
&#3627409360;&#3627409366;&#3627409367;
)=
&#3627409364;&#3627409362;
&#3627409360;&#3627409366;&#3627409367;
⇒ ??????&#3627408424;&#3627408428; ??????=+√
&#3627409364;&#3627409362;
&#3627409360;&#3627409366;&#3627409367;
=
&#3627409366;
&#3627409359;&#3627409365;
.
And, &#3627408428;??????&#3627408423;
&#3627409360;
??????=(&#3627409359;−??????&#3627408424;&#3627408428;
&#3627409360;
??????)=(&#3627409359;−
&#3627409359;&#3627409362;&#3627409362;
&#3627409359;&#3627409364;&#3627409367;
)=
&#3627409360;&#3627409363;
&#3627409359;&#3627409364;&#3627409367;
⇒ &#3627408428;??????&#3627408423; ??????=+√
&#3627409360;&#3627409363;
&#3627409359;&#3627409364;&#3627409367;
=
&#3627409363;
&#3627409359;&#3627409361;
.
&#3627408429;??????&#3627408423; ??????=
&#3627408428;??????&#3627408527; ??????
??????&#3627408424;&#3627408428; ??????
=(
&#3627409359;&#3627409363;
&#3627409359;&#3627409365;
×
&#3627409359;&#3627409365;
&#3627409366;
)=
&#3627409359;&#3627409363;
&#3627409366;
and &#3627408429;??????&#3627408423; ??????=
&#3627408428;??????&#3627408423; ??????
??????&#3627408424;&#3627408428; ??????
=(
&#3627409363;
&#3627409359;&#3627409361;
×
&#3627409359;&#3627409361;
&#3627409359;&#3627409360;
)=
&#3627409363;
&#3627409359;&#3627409360;
.

Maths Learning Centre, Jalandhar Maths Learning Centre, Jalandhar Maths Learning Centre, Jalandhar
Maths Learning Centre, Jalandhar Maths Learning Centre, Jalandhar Maths Learning Centre, Jalandhar
Maths Learning Centre, Jalandhar Maths Learning Centre, Jalandhar Maths Learning Centre, Jalandhar
Maths Learning Centre, Jalandhar Maths Learning Centre, Jalandhar Maths Learning Centre, Jalandhar
Maths Learning Centre, Jalandhar Maths Learning Centre, Jalandhar Maths Learning Centre, Jalandhar
Maths Learning Centre, Jalandhar Maths Learning Centre, Jalandhar Maths Learning Centre, Jalandhar
Maths Learning Centre, Jalandhar Maths Learning Centre, Jalandhar Maths Learning Centre, Jalandhar

youtube.com/@MathematicsOnlineLectures
Now, (i) &#3627408428;??????&#3627408423; (??????+??????)= &#3627408428;??????&#3627408423; ?????? ??????&#3627408424;&#3627408428; ??????+ ??????&#3627408424;&#3627408428; ?????? &#3627408428;??????&#3627408423; ??????
=(
&#3627409359;&#3627409363;
&#3627409359;&#3627409365;
×
&#3627409359;&#3627409360;
&#3627409359;&#3627409361;
)+(
&#3627409366;
&#3627409359;&#3627409365;
×
&#3627409363;
&#3627409359;&#3627409361;
)=(
&#3627409359;&#3627409366;&#3627409358;
&#3627409360;&#3627409360;&#3627409359;
+
&#3627409362;&#3627409358;
&#3627409360;&#3627409360;&#3627409359;
)=
&#3627409360;&#3627409360;&#3627409358;
&#3627409360;&#3627409360;&#3627409359;
.

(ii) ??????&#3627408424;&#3627408428; (??????−??????)= ??????&#3627408424;&#3627408428; ?????? ??????&#3627408424;&#3627408428; ??????+ &#3627408428;??????&#3627408423; ?????? &#3627408428;??????&#3627408423; ??????
=(
&#3627409366;
&#3627409359;&#3627409365;
×
&#3627409359;&#3627409360;
&#3627409359;&#3627409361;
)+(
&#3627409359;&#3627409363;
&#3627409359;&#3627409365;
×
&#3627409363;
&#3627409359;&#3627409361;
)=(
&#3627409367;&#3627409364;
&#3627409360;&#3627409360;&#3627409359;
+
&#3627409365;&#3627409363;
&#3627409360;&#3627409360;&#3627409359;
)=
&#3627409359;&#3627409365;&#3627409359;
&#3627409360;&#3627409360;&#3627409359;
.

(iii) &#3627408429;??????&#3627408423; (??????+??????)=
&#3627408429;??????&#3627408423; ??????+ &#3627408429;??????&#3627408423; ??????
&#3627409359;− &#3627408429;??????&#3627408423; ?????? &#3627408429;??????&#3627408423; ??????
=
(
&#3627409359;&#3627409363;
&#3627409366;
+
&#3627409363;
&#3627409359;&#3627409360;
)
{&#3627409359;−(
&#3627409359;&#3627409363;
&#3627409366;
×
&#3627409363;
&#3627409359;&#3627409360;
)}
=
(
&#3627409363;&#3627409363;
&#3627409360;&#3627409362;
)
(&#3627409359;−
&#3627409360;&#3627409363;
&#3627409361;&#3627409360;
)
=(
&#3627409363;&#3627409363;
&#3627409360;&#3627409362;
×
&#3627409361;&#3627409360;
&#3627409365;
)=
&#3627409360;&#3627409360;&#3627409358;
&#3627409360;&#3627409359;
.

Maths Learning Centre, Jalandhar Maths Learning Centre, Jalandhar Maths Learning Centre, Jalandhar
Maths Learning Centre, Jalandhar Maths Learning Centre, Jalandhar Maths Learning Centre, Jalandhar
Maths Learning Centre, Jalandhar Maths Learning Centre, Jalandhar Maths Learning Centre, Jalandhar
Maths Learning Centre, Jalandhar Maths Learning Centre, Jalandhar Maths Learning Centre, Jalandhar
Maths Learning Centre, Jalandhar Maths Learning Centre, Jalandhar Maths Learning Centre, Jalandhar
Maths Learning Centre, Jalandhar Maths Learning Centre, Jalandhar Maths Learning Centre, Jalandhar
Maths Learning Centre, Jalandhar Maths Learning Centre, Jalandhar Maths Learning Centre, Jalandhar

youtube.com/@MathematicsOnlineLectures
If &#3627408428;??????&#3627408423; ??????=
&#3627409361;
&#3627409363;
and ??????&#3627408424;&#3627408428; ??????=
−&#3627409359;&#3627409360;
&#3627409359;&#3627409361;
, where ?????? and ?????? both lie in the second quadrant, find the
values of (i) &#3627408428;??????&#3627408423; (??????−??????) , (ii) ??????&#3627408424;&#3627408428; (??????+??????) , (iii) &#3627408429;??????&#3627408423; (??????−??????) .
Given: &#3627408428;??????&#3627408423; ??????=
&#3627409361;
&#3627409363;
and ??????&#3627408424;&#3627408428; ??????=
−&#3627409359;&#3627409360;
&#3627409359;&#3627409361;
.
Since ?????? lies in the second quadrant, we have &#3627408428;??????&#3627408423; ??????>&#3627409358;, ??????&#3627408424;&#3627408428; ??????<&#3627409358; and &#3627408429;??????&#3627408423; ??????<&#3627409358;.
Again, since ??????lies in the second quadrant, we have &#3627408428;??????&#3627408423; ??????>&#3627409358;, ??????&#3627408424;&#3627408428; ??????<&#3627409358; and
&#3627408429;??????&#3627408423; ??????<&#3627409358;.
Now, ??????&#3627408424;&#3627408428; ??????=−√&#3627409359;−&#3627408428;??????&#3627408423;
&#3627409360;
??????=−√&#3627409359;−
&#3627409367;
&#3627409360;&#3627409363;
=−√
&#3627409359;&#3627409364;
&#3627409360;&#3627409363;
=
−&#3627409362;
&#3627409363;

and &#3627408429;??????&#3627408423; ??????=
&#3627408428;??????&#3627408423; ??????
??????&#3627408424;&#3627408428; ??????
=
&#3627409361;
&#3627409363;
×(−
&#3627409363;
&#3627409362;
)=
−&#3627409361;
&#3627409362;
.
Also, &#3627408428;??????&#3627408423; ??????=+√&#3627409359;−??????&#3627408424;&#3627408428;
&#3627409360;
??????=+√&#3627409359;−
&#3627409359;&#3627409362;&#3627409362;
&#3627409359;&#3627409364;&#3627409367;
=+√
&#3627409360;&#3627409363;
&#3627409359;&#3627409364;&#3627409367;
=+
&#3627409363;
&#3627409359;&#3627409361;

and &#3627408429;??????&#3627408423; ??????=
&#3627408428;??????&#3627408423; ??????
??????&#3627408424;&#3627408428; ??????
=
&#3627409363;
&#3627409359;&#3627409361;
×(
−&#3627409359;&#3627409361;
&#3627409359;&#3627409360;
)=
−&#3627409363;
&#3627409359;&#3627409360;
.

Maths Learning Centre, Jalandhar Maths Learning Centre, Jalandhar Maths Learning Centre, Jalandhar
Maths Learning Centre, Jalandhar Maths Learning Centre, Jalandhar Maths Learning Centre, Jalandhar
Maths Learning Centre, Jalandhar Maths Learning Centre, Jalandhar Maths Learning Centre, Jalandhar
Maths Learning Centre, Jalandhar Maths Learning Centre, Jalandhar Maths Learning Centre, Jalandhar
Maths Learning Centre, Jalandhar Maths Learning Centre, Jalandhar Maths Learning Centre, Jalandhar
Maths Learning Centre, Jalandhar Maths Learning Centre, Jalandhar Maths Learning Centre, Jalandhar
Maths Learning Centre, Jalandhar Maths Learning Centre, Jalandhar Maths Learning Centre, Jalandhar

youtube.com/@MathematicsOnlineLectures
(i) &#3627408428;??????&#3627408423; (??????−??????)=&#3627408428;??????&#3627408527; ?????? ??????&#3627408424;&#3627408428; ??????− ??????&#3627408424;&#3627408428; ?????? &#3627408428;??????&#3627408423; ??????
={
&#3627409361;
&#3627409363;
×(
−&#3627409359;&#3627409360;
&#3627409359;&#3627409361;
)}−{(
−&#3627409362;
&#3627409363;
)×
&#3627409363;
&#3627409359;&#3627409361;
}=(
−&#3627409361;&#3627409364;
&#3627409364;&#3627409363;
+
&#3627409360;&#3627409358;
&#3627409364;&#3627409363;
)=
−&#3627409359;&#3627409364;
&#3627409364;&#3627409363;
.

(ii) ??????&#3627408424;&#3627408428; (??????+??????)= ??????&#3627408424;&#3627408428; ?????? ??????&#3627408424;&#3627408428; ??????− &#3627408428;??????&#3627408423; ?????? &#3627408428;??????&#3627408423; ??????
={(
−&#3627409362;
&#3627409363;
)×(
−&#3627409359;&#3627409360;
&#3627409359;&#3627409361;
)}−{
&#3627409361;
&#3627409363;
×
&#3627409363;
&#3627409359;&#3627409361;
}=(
&#3627409362;&#3627409366;
&#3627409364;&#3627409363;
−
&#3627409361;
&#3627409359;&#3627409361;
)=
&#3627409361;&#3627409361;
&#3627409364;&#3627409363;
.

(iii) &#3627408429;??????&#3627408423; (??????−??????)=
&#3627408429;??????&#3627408423; ??????− &#3627408429;??????&#3627408423; ??????
&#3627409359;+ &#3627408429;??????&#3627408423; ?????? &#3627408429;??????&#3627408423; ??????
=
(
−&#3627409361;
&#3627409362;
)−(
−&#3627409363;
&#3627409359;&#3627409360;
)
&#3627409359;+{(
−&#3627409361;
&#3627409362;
)×(
−&#3627409363;
&#3627409359;&#3627409360;
)}
=
(
−&#3627409361;
&#3627409362;
+
&#3627409363;
&#3627409359;&#3627409360;
)
(&#3627409359;+
&#3627409363;
&#3627409359;&#3627409364;
)
=(
−&#3627409359;
&#3627409361;
)×
&#3627409359;&#3627409364;
&#3627409360;&#3627409359;
=
−&#3627409359;&#3627409364;
&#3627409364;&#3627409361;
.

Maths Learning Centre, Jalandhar Maths Learning Centre, Jalandhar Maths Learning Centre, Jalandhar
Maths Learning Centre, Jalandhar Maths Learning Centre, Jalandhar Maths Learning Centre, Jalandhar
Maths Learning Centre, Jalandhar Maths Learning Centre, Jalandhar Maths Learning Centre, Jalandhar
Maths Learning Centre, Jalandhar Maths Learning Centre, Jalandhar Maths Learning Centre, Jalandhar
Maths Learning Centre, Jalandhar Maths Learning Centre, Jalandhar Maths Learning Centre, Jalandhar
Maths Learning Centre, Jalandhar Maths Learning Centre, Jalandhar Maths Learning Centre, Jalandhar
Maths Learning Centre, Jalandhar Maths Learning Centre, Jalandhar Maths Learning Centre, Jalandhar

youtube.com/@MathematicsOnlineLectures
If ??????&#3627408424;&#3627408428; ??????=
&#3627409362;
&#3627409363;
and ??????&#3627408424;&#3627408428; ??????=
&#3627409359;&#3627409360;
&#3627409359;&#3627409361;
, where 6 and ?????? both lie in the fourth quadrant, find the
values of (i) ??????&#3627408424;&#3627408428; (??????+??????) , (ii) &#3627408428;??????&#3627408423; (??????−??????) , (iii) &#3627408429;??????&#3627408423; (??????+??????) .
Given: ??????&#3627408424;&#3627408428; ??????=
&#3627409362;
&#3627409363;
and ??????&#3627408424;&#3627408428; ??????=
&#3627409359;&#3627409360;
&#3627409359;&#3627409361;
.
Since ?????? lies in the fourth quadrant, we have ??????&#3627408424;&#3627408428; ??????>&#3627409358;, &#3627408428;i&#3627408423;??????<&#3627409358; and &#3627408429;??????&#3627408423; ??????<&#3627409358;.
Again, since ??????lies in the fourth quadrantJ we have ??????&#3627408424;&#3627408428; ??????>&#3627409358;, &#3627408428;??????&#3627408423; ??????<&#3627409358; and
&#3627408429;??????&#3627408423; ??????<&#3627409358;.
Now, &#3627408428;??????&#3627408423; ??????=−√&#3627409359;−??????&#3627408424;&#3627408428;
&#3627409360;
??????=−√&#3627409359;−
&#3627409359;&#3627409364;
&#3627409360;&#3627409363;
=−√
&#3627409367;
&#3627409360;&#3627409363;
=
−&#3627409361;
&#3627409363;

and &#3627408429;??????&#3627408423; ??????=
&#3627408428;??????&#3627408423; ??????
??????&#3627408424;&#3627408428; ??????
=(
−&#3627409361;
&#3627409363;
)×
&#3627409363;
&#3627409362;
=
−&#3627409361;
&#3627409362;
.
Also, &#3627408428;??????&#3627408423; ??????=−√&#3627409359;−??????&#3627408424;&#3627408428;
&#3627409360;
??????=−√&#3627409359;−
&#3627409359;&#3627409362;&#3627409362;
&#3627409359;&#3627409364;&#3627409367;
=−√
&#3627409360;&#3627409363;
&#3627409359;&#3627409364;&#3627409367;
=
−&#3627409363;
&#3627409359;&#3627409361;

and &#3627408429;??????&#3627408423; ??????=
&#3627408428;??????&#3627408423; ??????
??????&#3627408424;&#3627408428; ??????
=(
−&#3627409363;
&#3627409359;&#3627409361;
)×
&#3627409359;&#3627409361;
&#3627409359;&#3627409360;
=
−&#3627409363;
&#3627409359;&#3627409360;
.

Maths Learning Centre, Jalandhar Maths Learning Centre, Jalandhar Maths Learning Centre, Jalandhar
Maths Learning Centre, Jalandhar Maths Learning Centre, Jalandhar Maths Learning Centre, Jalandhar
Maths Learning Centre, Jalandhar Maths Learning Centre, Jalandhar Maths Learning Centre, Jalandhar
Maths Learning Centre, Jalandhar Maths Learning Centre, Jalandhar Maths Learning Centre, Jalandhar
Maths Learning Centre, Jalandhar Maths Learning Centre, Jalandhar Maths Learning Centre, Jalandhar
Maths Learning Centre, Jalandhar Maths Learning Centre, Jalandhar Maths Learning Centre, Jalandhar
Maths Learning Centre, Jalandhar Maths Learning Centre, Jalandhar Maths Learning Centre, Jalandhar

youtube.com/@MathematicsOnlineLectures
(i) ??????&#3627408424;&#3627408428; (??????+??????)= ??????&#3627408424;&#3627408428; ?????? ??????&#3627408424;&#3627408428; ??????− &#3627408428;??????&#3627408423; ??????&#3627408428;??????&#3627408423;??????
=(
&#3627409362;
&#3627409363;
×
&#3627409359;&#3627409360;
&#3627409359;&#3627409361;
)−{(
−&#3627409361;
&#3627409363;
)×(
−&#3627409363;
&#3627409359;&#3627409361;
)}=(
&#3627409362;&#3627409366;
&#3627409364;&#3627409363;
−
&#3627409359;&#3627409363;
&#3627409364;&#3627409363;
)=
&#3627409361;&#3627409361;
&#3627409364;&#3627409363;
.

(ii) &#3627408428;??????&#3627408423; (??????−??????)= &#3627408428;??????&#3627408423; ?????? ??????&#3627408424;&#3627408428; ??????− ??????&#3627408424;&#3627408428; ?????? &#3627408428;??????&#3627408423; ??????
={(
−&#3627409361;
&#3627409363;
)×(
&#3627409359;&#3627409360;
&#3627409359;&#3627409361;
)}−{
&#3627409362;
&#3627409363;
×(
−&#3627409363;
&#3627409359;&#3627409361;
)}=(
−&#3627409361;&#3627409364;
&#3627409364;&#3627409363;
+
&#3627409360;&#3627409358;
&#3627409364;&#3627409363;
)=
−&#3627409359;&#3627409364;
&#3627409364;&#3627409363;
.

(iii) &#3627408429;??????&#3627408423; (??????+??????)=
&#3627408429;??????&#3627408423; ??????+ &#3627408429;??????&#3627408423; ??????
&#3627409359;− &#3627408429;??????&#3627408423; ?????? &#3627408429;??????&#3627408423; ??????

=
(
−&#3627409361;
&#3627409362;
)+(
−&#3627409363;
&#3627409359;&#3627409360;
)
&#3627409359;−{(
−&#3627409361;
&#3627409362;
)×(
−&#3627409363;
&#3627409359;&#3627409360;
)}
=
(
−&#3627409365;
&#3627409364;
)
(&#3627409359;−
&#3627409363;
&#3627409359;&#3627409364;
)
=(
−&#3627409365;
&#3627409364;
)×
&#3627409359;&#3627409364;
&#3627409359;&#3627409359;
=
−&#3627409363;&#3627409364;
&#3627409361;&#3627409361;
.
Maths Learning Centre, Jalandhar Maths Learning Centre, Jalandhar Maths Learning Centre, Jalandhar
Maths Learning Centre, Jalandhar Maths Learning Centre, Jalandhar Maths Learning Centre, Jalandhar
Maths Learning Centre, Jalandhar Maths Learning Centre, Jalandhar Maths Learning Centre, Jalandhar
Maths Learning Centre, Jalandhar Maths Learning Centre, Jalandhar Maths Learning Centre, Jalandhar
Maths Learning Centre, Jalandhar Maths Learning Centre, Jalandhar Maths Learning Centre, Jalandhar
Maths Learning Centre, Jalandhar Maths Learning Centre, Jalandhar Maths Learning Centre, Jalandhar
Maths Learning Centre, Jalandhar Maths Learning Centre, Jalandhar Maths Learning Centre, Jalandhar

youtube.com/@MathematicsOnlineLectures
If ??????&#3627408424;&#3627408429; &#3627409206;=
&#3627409359;
&#3627409360;
and &#3627408428;???????????? &#3627409207;=
−&#3627409363;
&#3627409361;
, where &#3627409206;∈(??????,
&#3627409361;??????
&#3627409360;
)and &#3627409207;∈(
??????
&#3627409360;
, ??????), find the value of
&#3627408429;??????&#3627408423; (&#3627409206;+&#3627409207;) .
Here, &#3627409206; lies in Quadrant III and, therefore, &#3627408429;??????&#3627408423; &#3627409206; is positive.
Now, ??????&#3627408424;&#3627408429; &#3627409206;=
&#3627409359;
&#3627409360;
⇒ &#3627408429;??????&#3627408423; &#3627409206;=&#3627409360;.
Again, &#3627409207; lies in Quadrant II and, therefore, &#3627408428;??????&#3627408423; &#3627409207; is positive and ??????&#3627408424;&#3627408428; &#3627409207; is negative.
Now, &#3627408428;???????????? &#3627409207;=
−&#3627409363;
&#3627409361;
⇒ ??????&#3627408424;&#3627408428; &#3627409207;=
−&#3627409361;
&#3627409363;
.
And, &#3627408428;??????&#3627408423; &#3627409207;=+√&#3627409359;−??????&#3627408424;&#3627408428;
&#3627409360;
&#3627409207;=+√&#3627409359;−
&#3627409367;
&#3627409360;&#3627409363;
=+√
&#3627409359;&#3627409364;
&#3627409360;&#3627409363;
=
&#3627409362;
&#3627409363;
.
&#3627408429;??????&#3627408423; &#3627409207;=
&#3627408428;??????&#3627408423; &#3627409207;
??????&#3627408424;&#3627408428; &#3627409207;
=
&#3627409362;
&#3627409363;
×(
−&#3627409363;
&#3627409361;
)=
−&#3627409362;
&#3627409361;
.
&#3627408429;??????&#3627408423; (&#3627409206;+&#3627409207;)=
&#3627408429;??????&#3627408423; &#3627409206;+ &#3627408429;??????&#3627408423; &#3627409207;
&#3627409359;− &#3627408429;??????&#3627408423; &#3627409206; &#3627408429;??????&#3627408423; &#3627409207;
=
&#3627409360;−
&#3627409362;
&#3627409361;
&#3627409359;−&#3627409360;×(
−&#3627409362;
&#3627409361;
)
=
(
&#3627409360;
&#3627409361;
)
(
&#3627409359;&#3627409359;
&#3627409361;
)
=(
&#3627409360;
&#3627409361;
×
&#3627409361;
&#3627409359;&#3627409359;
)=
&#3627409360;
&#3627409359;&#3627409359;
.

Maths Learning Centre, Jalandhar Maths Learning Centre, Jalandhar Maths Learning Centre, Jalandhar
Maths Learning Centre, Jalandhar Maths Learning Centre, Jalandhar Maths Learning Centre, Jalandhar
Maths Learning Centre, Jalandhar Maths Learning Centre, Jalandhar Maths Learning Centre, Jalandhar
Maths Learning Centre, Jalandhar Maths Learning Centre, Jalandhar Maths Learning Centre, Jalandhar
Maths Learning Centre, Jalandhar Maths Learning Centre, Jalandhar Maths Learning Centre, Jalandhar
Maths Learning Centre, Jalandhar Maths Learning Centre, Jalandhar Maths Learning Centre, Jalandhar
Maths Learning Centre, Jalandhar Maths Learning Centre, Jalandhar Maths Learning Centre, Jalandhar

youtube.com/@MathematicsOnlineLectures
Prove that &#3627408429;??????&#3627408423; &#3627409363;&#3627409364;
∘
=
??????&#3627408424;&#3627408428; &#3627409359;&#3627409359;
∘
+ &#3627408428;??????&#3627408423; &#3627409359;&#3627409359;
∘
??????&#3627408424;&#3627408428; &#3627409359;&#3627409359;
∘
− &#3627408428;??????&#3627408423; &#3627409359;&#3627409359;
∘
.
We have ????????????&#3627408402;= &#3627408429;??????&#3627408423; &#3627409363;&#3627409364;
∘
= &#3627408429;??????&#3627408423; (&#3627409362;&#3627409363;
∘
+&#3627409359;&#3627409359;
∘
)=
&#3627408429;??????&#3627408423; &#3627409362;&#3627409363;
∘
+ &#3627408429;??????&#3627408423; &#3627409359;&#3627409359;
∘
&#3627409359;− &#3627408429;??????&#3627408423; &#3627409362;&#3627409363;
∘
&#3627408429;??????&#3627408423; &#3627409359;&#3627409359;
∘

=
&#3627409359;+ &#3627408429;??????&#3627408423; &#3627409359;&#3627409359;
∘
&#3627409359;− &#3627408429;??????&#3627408423; &#3627409359;&#3627409359;
∘
=
&#3627409359;+
&#3627408428;&#3627408422;̇&#3627409359;&#3627409359;
∘
??????&#3627408424;&#3627408428; &#3627409359;&#3627409359;
∘
&#3627409359;−
&#3627408428;??????&#3627408423; &#3627409359;&#3627409359;
∘
??????&#3627408424;&#3627408428; &#3627409359;&#3627409359;
∘
=
??????&#3627408424;&#3627408428; &#3627409359;&#3627409359;
∘
+ &#3627408428;??????&#3627408423; &#3627409359;&#3627409359;
∘
??????&#3627408424;&#3627408428; &#3627409359;&#3627409359;
∘
− &#3627408428;??????&#3627408423; &#3627409359;&#3627409359;
∘
=&#3627408401;??????&#3627408402;.

Prove that &#3627408429;??????&#3627408423; &#3627409365;&#3627409358;
∘
= &#3627408429;??????&#3627408423; &#3627409360;&#3627409358;
∘
+&#3627409360; &#3627408429;??????&#3627408423; &#3627409363;&#3627409358;
∘
.
We have &#3627408429;??????&#3627408423; &#3627409365;&#3627409358;
∘
= &#3627408429;??????&#3627408423; (&#3627409360;&#3627409358;
∘
+&#3627409363;&#3627409358;
∘
)⇒ &#3627408429;??????&#3627408423; &#3627409365;&#3627409358;
∘
=
&#3627408429;??????&#3627408423; &#3627409360;&#3627409358;
∘
+ &#3627408429;??????&#3627408423; &#3627409363;&#3627409358;
∘
&#3627409359;− &#3627408429;??????&#3627408423; &#3627409360;&#3627409358;
∘
&#3627408429;??????&#3627408423; &#3627409363;&#3627409358;
∘

⇒ &#3627408429;??????&#3627408423; &#3627409365;&#3627409358;
∘
− &#3627408429;??????&#3627408423; &#3627409365;&#3627409358;
∘
&#3627408429;??????&#3627408423; &#3627409360;&#3627409358;
∘
&#3627408429;??????&#3627408423; &#3627409363;&#3627409358;
∘
= &#3627408429;??????&#3627408423; &#3627409360;&#3627409358;
∘
+ &#3627408429;??????&#3627408423; &#3627409363;&#3627409358;
∘

⇒ &#3627408429;??????&#3627408423; &#3627409365;&#3627409358;
∘
− &#3627408429;??????&#3627408423; (&#3627409367;&#3627409358;
∘
−&#3627409360;&#3627409358;
∘
) &#3627408429;??????&#3627408423; &#3627409360;&#3627409358;
∘
&#3627408429;??????&#3627408423; &#3627409363;&#3627409358;
∘
= &#3627408429;??????&#3627408423; &#3627409360;&#3627409358;
∘
+ &#3627408429;??????&#3627408423; &#3627409363;&#3627409358;
∘

⇒ &#3627408429;??????&#3627408423; &#3627409365;&#3627409358;
∘
− ??????&#3627408424;&#3627408429; &#3627409360;&#3627409358;
∘
&#3627408429;??????&#3627408423; &#3627409360;&#3627409358;
∘
&#3627408429;??????&#3627408423; &#3627409363;&#3627409358;
∘
= &#3627408429;??????&#3627408423; &#3627409360;&#3627409358;
∘
+ &#3627408429;??????&#3627408423; &#3627409363;&#3627409358;
∘

⇒ &#3627408429;??????&#3627408423; &#3627409365;&#3627409358;
∘
− &#3627408429;??????&#3627408423; &#3627409363;&#3627409358;
∘
= &#3627408429;??????&#3627408423; &#3627409360;&#3627409358;
∘
+ &#3627408429;??????&#3627408423; &#3627409363;&#3627409358;
∘

⇒ &#3627408429;??????&#3627408423; &#3627409365;&#3627409358;
∘
= &#3627408429;??????&#3627408423; &#3627409360;&#3627409358;
∘
+&#3627409360; &#3627408429;??????&#3627408423; &#3627409363;&#3627409358;
∘
.

Maths Learning Centre, Jalandhar Maths Learning Centre, Jalandhar Maths Learning Centre, Jalandhar
Maths Learning Centre, Jalandhar Maths Learning Centre, Jalandhar Maths Learning Centre, Jalandhar
Maths Learning Centre, Jalandhar Maths Learning Centre, Jalandhar Maths Learning Centre, Jalandhar
Maths Learning Centre, Jalandhar Maths Learning Centre, Jalandhar Maths Learning Centre, Jalandhar
Maths Learning Centre, Jalandhar Maths Learning Centre, Jalandhar Maths Learning Centre, Jalandhar
Maths Learning Centre, Jalandhar Maths Learning Centre, Jalandhar Maths Learning Centre, Jalandhar
Maths Learning Centre, Jalandhar Maths Learning Centre, Jalandhar Maths Learning Centre, Jalandhar

youtube.com/@MathematicsOnlineLectures
Show that ??????&#3627408424;&#3627408429; &#3627409360;&#3627408537; ??????&#3627408424;&#3627408429; &#3627408537;− ??????&#3627408424;&#3627408429; &#3627409361;&#3627408537; ??????&#3627408424;&#3627408429; &#3627409360;&#3627408537;− ??????&#3627408424;&#3627408429; &#3627409361;&#3627408537; ??????&#3627408424;&#3627408429; &#3627408537;=&#3627409359;.
We have
??????&#3627408424;&#3627408429; &#3627409361;&#3627408537;= ??????&#3627408424;&#3627408429; (&#3627409360;&#3627408537;+&#3627408537;)⇔ ??????&#3627408424;&#3627408429; &#3627409361;&#3627408537;=
??????&#3627408424;&#3627408429; &#3627409360;&#3627408537; ??????&#3627408424;&#3627408429; &#3627408537;−&#3627409359;
??????&#3627408424;&#3627408429; &#3627409360;&#3627408537;+ ??????&#3627408424;&#3627408429; &#3627408537;

⇔ ??????&#3627408424;&#3627408429; &#3627409361;&#3627408537; ??????&#3627408424;&#3627408429; &#3627409360;&#3627408537;+ ??????&#3627408424;&#3627408429; &#3627409361;&#3627408537; ??????&#3627408424;&#3627408429; &#3627408537;= ??????&#3627408424;&#3627408429; &#3627409360;&#3627408537; ??????&#3627408424;&#3627408429; &#3627408537;−&#3627409359;
⇔ ??????&#3627408424;&#3627408429; &#3627409360;&#3627408537; ??????&#3627408424;&#3627408429; &#3627408537;− ??????&#3627408424;&#3627408429; &#3627409361;&#3627408537; ??????&#3627408424;&#3627408429; &#3627409360;&#3627408537;− ??????&#3627408424;&#3627408429; &#3627409361;&#3627408537; ??????&#3627408424;&#3627408429; &#3627408537;=&#3627409359;.

Maths Learning Centre, Jalandhar Maths Learning Centre, Jalandhar Maths Learning Centre, Jalandhar
Maths Learning Centre, Jalandhar Maths Learning Centre, Jalandhar Maths Learning Centre, Jalandhar
Maths Learning Centre, Jalandhar Maths Learning Centre, Jalandhar Maths Learning Centre, Jalandhar
Maths Learning Centre, Jalandhar Maths Learning Centre, Jalandhar Maths Learning Centre, Jalandhar
Maths Learning Centre, Jalandhar Maths Learning Centre, Jalandhar Maths Learning Centre, Jalandhar
Maths Learning Centre, Jalandhar Maths Learning Centre, Jalandhar Maths Learning Centre, Jalandhar
Maths Learning Centre, Jalandhar Maths Learning Centre, Jalandhar Maths Learning Centre, Jalandhar

Class 11_Chapter 3_Lecture_3p.pdf

About This Presentation

Slide Content

Tags

Categories

Download

Quick Actions

Statistics

Related Slideshows

Class 11_Chapter 3_Lecture_3p.pdf

About This Presentation

Slide Content

Slide 1

Slide 2

Slide 3

Slide 4

Slide 5

Slide 6

Slide 7

Slide 8

Slide 9

Slide 10

Slide 11

Slide 12

Slide 13

Slide 14

Slide 15

Slide 16

Slide 17

Slide 18

Slide 19

Slide 20

Slide 21

Slide 22

Slide 23

Slide 24

Tags

Categories

Download

Quick Actions

Statistics

Related Slideshows

Pray For The Peace Of Jerusalem and You Will Prosper

Don_t_Waste_Your_Life_God.....powerpoint

VILLASUR_FACTORS_TO_CONSIDER_IN_PLATING_SALAD_10-13.pdf

Fertility awareness methods for women in the society

Chapter 5 Arithmetic Functions Computer Organisation and Architecture

syakira bhasa inggris (1) (1).pptx.......