Trigonometry for class xi

1. WEIGHTAGE FOR CLASS ---XI



One Paper Three Hours Max Marks. 100

Units Marks
I. SETS AND FUNCTIONS

II. ALGEBRA

III. COORDINATE GEOMETRY

IV. CALCULUS

V. MATHEMATICAL REASONING

VI. STATISTICS AND PROBABILITY
29

37

13

06

03

12
100

Fundamental Trigonometric Identities
Before we start to prove trigonometric identities, we see
where the basic identities come from.
Recall the definitions of the reciprocal trigonometric
functions, csc θ, sec θ and cot θ
from the trigonometric functions chapter:

2. After we revise the fundamental identities, we learn
about:
Proving trigonometric identities

Now, consider the following diagram where the point (x,
y) defines an angle θ at the origin,
and the distance from the origin to the point is r units:

From the diagram, we can see that the ratios sin θ
and cos θ are defined as:

and

Now, we use these results to find an important definition
for tan θ:

3. Now, also so we can conclude that:


Also, for the values in the diagram, we can use
Pythagoras' Theorem and obtain:
y
2
+ x
2
= r
2

Dividing through by r
2
gives us:

so we obtain the important result:

4. sin2 θ + cos2 θ = 1
We now proceed to derive two other related
formulas that can be used when proving trigonometric
identities.
It is suggested that you remember how to find the
identities, rather than try to memorise each one.
Dividing sin
2
θ + cos
2
θ = 1 through by cos
2
θ gives us:

so
tan
2
θ + 1 = sec
2
θ

Dividing sin
2
θ + cos
2
θ = 1 through by sin
2
θ gives us:

so
1 + cot
2
θ = csc
2
θ
Trigonometric Identities Summary

5. Proving Trigonometric Identities
Suggestions...
1. Learn well the formulas given above (or at least,
know how to find them quickly).
The better you know the basic identities, the easier it
will be to recognise what is going on in the problems.
2. Work on the most complex side and simplify it so that
it has the same form as the simplest side.
Don't assume the identity to prove the identity.
This means don't work on both sides of the equals
side and try to meet in the middle.
3. Start on one side and make it look like the other
side.
4. Many of these come out quite easily if you express
everything on the most complex side in terms of
sine and cosine only.
5. In most examples where you see power 2 (that is,
2
), it will involve using the identity sin
2
θ + cos
2
θ = 1
(or one of the other 2 formulas that we derived
above).

6. Using these suggestions, you can simplify and prove
expressions involving trigonometric identities.
Prove that
sin y + sin y cot
2
y = cosec y
Answer

7. Functio
n
Abbreviati
on
Descripti
on
Identities (using radians)
Sine sin


Cosine cos


Tangent tan (or tg)


Cotange
nt
cot (or ctg
or ctn)

8. Secant sec


Cosecan
t
csc
(or cosec)



Note that these values can easily be memorized in the
form

but the angles are not equally spaced.

9. The values for 15°, 54° and 75° are slightly more
complicated. [ by using formulas of sin(A-B),sin(A+B)
Similarly for cosine function & tan function.]



Special values in trigonometric functions
There are some commonly used special values in
trigonometric functions, as shown in the following table.
Function







s
in
0





1
cos 1





0
tan 0


1



cot



1


0
sec 1



2


csc


2



1

10. The sine and cosine functions graphed on the Cartesian
plane.
For angles greater than 2π or less than −2π, simply
continue to rotate
around the circle; sine and cosine are periodic functions
with period 2π:


for any angle θ and any integer k.
The smallest positive period of a periodic function is
called the
primitive period of the function.
The primitive period of the sine or cosine is a full circle,
i.e.
2π radians or 360 degrees.

11. Figure 1
If Q(x,y) is the point on the circle where the string ends,
we may think of as being an angle by associating to it
the central angle with vertex O(0,0) and sides passing
through the points P and Q. If instead of wrapping a
length s of string around the unit circle, we decide to
wrap it around a circle of radius R, the angle (in radians)
generated in the process will satisfy the following
relation:

12. Observe that the length s of string gives the measure of
the angle only when R=1.
As a matter of common practice and convenience, it is
useful to measure angles in degrees, which are defined
by partitioning one whole revolution into 360 equal
parts, each of which is then called one degree. In this
way, one whole revolution around the unit circle
measures radians and also 360 degrees (or ), that
is:


Each degree may be further subdivided into 60 parts,
called minutes, and in turn each minute may be
subdivided into another 60 parts, called seconds:



Angle sum identities

13. Sine


Illustration of the sum formula.
Draw the angles α and β. Place P on the line defined by α
+ β at unit distance from the origin.
Let PQ be a perpendicular from P to the line defined by
the angle α. OQP is a right angle.
Let QA be a perpendicular from Q to the x axis, and PB
be a perpendicular from P to the x axis. OAQ is a right
angle.
Draw QR parallel to the x-axis. Now angle RPQ = α
(because

14. or PQ/OP = Sin
or OQ/OP = Cos (if OP ≠ 1)

, so
, so

Or = = = . + .
(∵ RB = AQ , PQ & OQ are the hyp. )
= . + . .
By substituting − β for β and using Symmetry, we also
get:

15. Cosine
Using the figure above,

OR PQ/OP = Sin
OR OQ/OP = Cos
, so

, so

16. Or = = = - .
(AB=RQ)
= . - . .
By substituting − β for β and using Symmetry, we also
get:


Also, using the complementary angle formulae,





**Another simple "proof" can be given using Euler's
formula known from complex analysis: Euler's
formula is:

Although it is more precise to say that Euler's
formula entails the trigonometric identities, it follows
that for angles α and β we have:

17. e
i(α + β)
= cos(α + β) + isin(α + β)
Also using the following properties of exponential
functions:
e
i(α + β)
= e
iα
e
iβ
= (cosα + isinα)(cosβ + isinβ)
Evaluating the product:
e
i(α + β)
= (cosαcosβ − sinαsinβ) + i(sinαcosβ +
sinβcosα)
This will only be equal to the previous expression we
got, if the imaginary and real parts are equal
respectively. Hence we get:
cos(α + β) = cosαcosβ − sinαsinβ
sin(α + β) = sinαcosβ + sinβcosα ]
Tangent and cotangent
From the sine and cosine formulae, we get


Dividing both numerator and denominator by cos α
cos β, we get

18. Similarly (using a division by sin α sin β), we get


Double-angle identities
From the angle sum identities, we get

and

The Pythagorean identities give the two alternative
forms for the latter of these:


The angle sum identities also give

19. **It can also be proved using Eulers Formula

Mulitplying the exponent by two yields

But replacing the angle with its doubled version,
which achieves the same result in the left side of the
equation, yields

It follows that

By multiplying we get

Because the imaginary and real parts have to be the
same, we are left with the original identities
Half-angle identities
The two identities giving alternative forms for cos 2θ
give these:

20. One must choose the sign of the square root
properly—note that if 2π is added to θ the quantities
inside the square roots are unchanged, but the left-
hand-sides of the equations change sign. Therefore
the correct sign to use depends on the value of θ.
tan function, we have

If we multiply the numerator and denominator inside
the square root by (1 + cos θ), and do a little
manipulation using the Pythagorean identities, we get

If instead we multiply the numerator and
denominator by (1 - cos θ), we get

This also gives

21. Similar manipulations for the cot function give

Example. verify the identity

Answer. We have

which gives

But

and since

and , we get finally

Remark. In general it is good to check whether the given
formula is correct. One way to do that is to substitute

22. some numbers for the variables. For example, if we take
a=b = 0, we get

or we may take . In this case we have

Example. Find the exact value of

Answer. We have

Hence, using the additions formulas for the cosine
function we get

Since

we get

23. Example. Find the exact value for

Answer. We have

Since

we get

Finally we have

Remark. Using the addition formulas, we generate the
following identities

24. Double-Angle and Half-Angle formulas are very useful.
For example, rational functions of sine and cosine wil be
very hard to integrate without these formulas. They are
as follow

Example. Check the identities

25. Answer. We will check the first one. the second one is
left to the reader as an exercise. We have

Hence

which implies

Many functions involving powers of sine and cosine are
hard to integrate. The use of Double-Angle formulas help
reduce the degree of difficulty.
Example. Write as an expression involving the
trigonometric functions with their first power.
Answer. We have

Hence

Since , we get

26. or

Example: Verify the identity

Answer. We have

Using the Double-Angle formulas we get

Putting stuff together we get

From the Double-Angle formulas, one may generate
easily the Half-Angle formulas

27. In particular, we have

Example. Use the Half-Angle formulas to find

Answer. Set . Then

Using the above formulas, we get

28. Since , then is a positive number.
Therefore, we have

Same arguments lead to

Example. Check the identities

Answer. First note that

which falls from the identity
. So we need to verify only one identity. For example, let
us verify that

using the Half-Angle formulas, we get

29. which reduces to

Product and Sum Formulas

From the Addition Formulas, we derive the following
trigonometric formulas (or identities)

Remark. It is clear that the third formula and the fourth
are identical (use the property to see it).
The above formulas are important whenever need rises to
transform the product of sine and cosine into a sum. This
is a very useful idea in techniques of integration.

30. Example. Express the product as a
sum of trigonometric functions.
Answer. We have

which gives

Note that the above formulas may be used to transform a
sum into a product via the identities

Example. Express as a product.
Answer. We have

31. Note that we used .
Example. Verify the formula

Answer. We have

and

Hence

which clearly implies

Example. Find the real number x such that
and

Answer. Many ways may be used to tackle this problem.
Let us use the above formulas. We have

32. Hence

Since , the equation gives
and the equation gives .
Therefore, the solutions to the equation

are

Example. Verify the identity

Answer. We have

Using the above formulas we get

Hence

33. which implies

Since , we get

TRIGONOMETRIC EQUATIONS


Example : Solve for x in the following equation.



There are an infinite number of solutions to this problem. To
solve for x, you must first isolate the tangent term.

34. = tan(± )
General solution:
X = nπ ± ∀ n Є Z(integers)
Example : Solve for x in the following equation.



There are an infinite number of solutions to this problem.
To solve for x, set the equation equal to zero and factor.

35. then

when
when , and when

when and
This is impossible because
The exact value solutions are and

36. Example : Solve for x in the following equation.



There are an infinite number of solutions to this problem.
Isolate the sine term. To do this, rewrite the left side of
the equation in an equivalent factored form.




The product of two factors equals zero if at least one of
the factors equals zeros. This means that
if or

We just transformed a difficult problem into two easier
problems. To find the solutions to the original equation,
, we find the
solutions to the equations OR

37. Sinx = sin(-π/6)
x
= nπ + (-1)
n
(π/6) ∀ nЄ Z
OR


sinx = sin(π/2)
x = nπ + (-1)
n
(π/2) ∀ nЄ Z.


Example : Solve for x in the following equation.
(general solution)



There are an infinite number of solutions to this problem.

38. Cos(3x-1) = cos(π/2)
3x-1 = 2nπ ± π/2
3x = 2nπ ± π/2 +1
x = 1/3(2nπ ± π/2 +1) ∀ n Є Z.


1. Solve the trigonometric equation analytically
4 tan x − sec
2
x = 0 (for 0 ≤ x < 2π)
Answer
4 tan x − sec
2
x = 0


In 0 ≤ x < 2π, we need to find values of 2x such that
0 ≤ 2x < 4π.

39. So

So

or x = 0.2618, 1.309, 3.403, 4.451
2. Solve the trigonometric equation analytically for 0
≤ x < 2π:
sin 2x cos x − cos 2x sin x = 0
Answer
We recognise the left hand side to be in the form:
sin(a − b) = sin a cos b − cos a sin b,
where a = 2x and b = x.
So
sin 2x cos x − cos 2x sin x
= sin(2x − x)
= sin x
Now, we know the solutions of sin x = 0 to be:

40. x = 0, π.
3. Solve the given trigonometric equation
analytically and by graphical method (for 0 ≤ x < 2π):
sin 4x − cos 2x = 0
Answer
sin 4x − cos 2x = 0
2sin 2x cos 2x − cos 2x = 0
cos2x (2sin 2x - 1) = 0

EITHER
cos 2x = 0


OR
sin 2x = 1/2

41. Question Solve the equation tan 2θ − cot 2θ = 0 for
0 ≤ θ < 2π.
Answer

tan
2
2θ = 1
tan 2θ = ± 1

Since 0 ≤ θ < 2π , we need to consider values of 2θ
such that 0 ≤ 2θ < 4π. Hence, solving the above
equation, we have:

So

42. Question Solve the equation
for 0 ≤ θ < 2π.
Answer
By using the half angle formula for and
then squaring both sides, we get:
= 1 + cos x

So we have:
2 cos
2
x + 3 cos x + 1 = 0
(2 cos x + 1)(cos x + 1) = 0
Solving, we get
cos x = − 0.5 or cos x = − 1

43. Now gives .

However, on checking in the original equation, we
note that

but

So the only solution for this part is

Also, cos x = − 1 gives x = π.
So the solutions for the equation are


GENERAL SOLUTIONS OF TRIGONOMETRIC
EQUATIONS
1. If sinx = 0 ⇨ x = nπ, ∀ n є Z

44. 2. If cosx = 0 ⇨ x = (2n+1) π/2, ∀ n Є Z
3. If tanx = 0 ⇨ x = nπ, n Є Z
4. If sinx = six ⇨ x = nπ + (-1)
n
, ∀ n Є Z
5. If cosx = cos ⇨ x = 2nπ ±, ∀ n Є Z
6. If tanx = tan ⇨ x = nπ+, ∀ n Є Z.

EXAMPLE: Solve the equation: sin3θ + cos2θ = 0
SOLUTION: cos2θ = - sin3θ ⇨ cos2θ = cos (
+3θ)
⇨ 2θ = 2nπ± (
+3θ) , ∀ nЄZ
-θ = nπ+ and
5θ = nπ- , ∀ nєz.


Question –1 If cos(A+B)=4/5 , sin(A-B)=5/13 and
A,B lie between 0 and π/4 , prove that
tan 2A = 56/33.
Answer: Since A-B Є (-π/4 , π/4) and A+B Є
(0,π/2) both are positive sin(A+B)=3/5 , cos(A-B)=12/13

45. tan(A+B)=3/4 , tan(A-B) = 5/12 then
tan2A = tan(A+B+A-B) = =56/33
Question – 2 If cos (A-B)+cos(B-C)+cos(C-A) = -3/2 ,
Prove that
CosA+cosB+cosC = sinA+sinB+sinC = 0
Answer : From given result we get

2cosAcosB+2sinAsinB+2cosBcosC+2sinBsinC+2cosAcosC
+2sinAsinC +3 =0

2cosAcosB+2sinAsinB+2cosBcosC+2sinBsinC+2cosAcosC
+2sinAsinC + sin
2
A+cos
2
A+sin
2
B+cos
2
B
+sin
2
C+cos
2
C = 0. ( 3=1+1+1 and 1 can be
written as sin
2
x+cos
2
x)
(sin
2
A+sin
2
B+sin
2
C+
2sinAsinB+2sinBsinC+2sinAsinC)
+(cos
2
A+cos
2
B+cos
2
C
+2cosAcosB+2cosBcosC+2cosAcosC) = 0
(sinA+sinB+sinC)
2
+ (cosA+cosB+cosC)
2
= 0.
Question – 3 Prove that (1+cos ) (1+ cos ) (1+
cos ) (1+ cos ) = .

46. Answer : cos = cos( π - ) = - cos , cos =
cos ( π - ) = - cos
L.H.S. (1+cos ) (1+ cos ) (1+ cos
) (1+ cos )
= (1 – cos
2
) (1 – cos
2
) = sin
2

. sin
2

= (2sin
2
). (2sin
2
) = (1 -
cos ) ( 1 - cos ) = .
Question – 4 Prove that cos cos cos cos =
.
Answer : L.H.S. ⇨ - cos cos cos cos
[∵cos = cos(π - ) ]
= - [ ] , where A=
[∵ all angels are in G.P. , short-cut Method]
= - [ ] = - =
= . [ ∵ sin (π + ) lies in 3
rd
quadrant]
OR

47. L.H.S. ⇨ - (2 sin cos ) cos
cos cos
= - ( 2sin cos )cos
cos
= - ( 2sin cos )cos

= - (2 sin .cos
) , now you can apply above result.
Similarly you can prove cosAcos2Acos4Acos8A =
sin16A/16sinA.

Question – 5 (i) Prove that sin20
0
sin40
0
sin60
0
sin80
0
=

Answer : L.H.S. sin20
0
sin40
0
sin60
0
sin80
0

(
⇨ (2sin20
0
sin40
0
sin80
0
)
⇨ [(cos20
0
– cos60
0
) sin80
0
]
⇨ [cos20
0
. sin80
0
– cos60
0
. sin80
0
]

48. (1/2)

⇨ [( cos20
0
.sin80
0
– sin80
0
]⇨
[(2 cos20
0
.sin80
0
–sin80
0
]
⇨ [(sin100
0
+sin60
0
– sin80
0
] =
( ∵ sin100
0
lies in 2
nd
quadrant)
[sin(180
0
- 80
0
)] = sin80
0

(ii) Prove that: cos cos cos = - .
[Hint: let x = , then (2sinx cosx cos2x cos4x)]
Sin2x
(iii) Prove that: tan20
0
tan40
0
tan80
0
= tan60
0
.
[hint: L.H.S. solve as above method.]

Question – 6 Solve : 2sinx + cosx = 1+ sinx
Answer : sinx + cosx = 1
cosx + sinx = [ dividing by
,where a = and b = 1]
cos . cosx + .sinx = cos

49. ⇨ cos(x - ) = cos ⇨ x - =
2nπ ± ∀ n Є Z (integers)
⇨ x = 2nπ ± + = 2nπ + ,
2nπ - ∀ n Є Z (integers).
Question – 7 Solve: 3cos
2
x - 2 sinx .cosx – 3sin
2
x =
0.
Answer: 3 cos
2
x - 3 sinx .cosx + sinx
.cosx – 3sin
2
x = 0
⇨ 3cosx ( cosx - sinx) + sinx
(cosx - sinx) = 0
⇨ (3cosx + sinx) (cosx -
sinx) = 0
⇨ (3cosx + sinx) = 0 or (cosx -
sinx) = 0
⇨ tanx = - = tan(- ) or
tanx = =tan( )
⇨ x = nπ+(- ) ∀ n Є Z
(integers) or x = nπ +

Question – 8 If , are the acute angles and cos2
= , show that tan = tan.

50. Answer: According to required result , we
have to convert given part into tangent function
By using cos2 =
∴ we will get =
=
=

By C & D =

⇨ =
2

⇨ = 2 ⇨
tan = tan.

Question – 9 Prove that =
cot4A.

51. Answer : =

[∵ sinA+sinB = 2sin(A+B)/2. cos(A-
B)/2 & cosA-cosB = - 2sin(A+B)/2.sin(A-B)/2 ]
=
= cot 4A.

Question – 10 Prove that
= 2COSθ.
Answer: =

=
=

= 2cosθ. [By using = 2cos
2
θ]

Question – 11 Find the general solution of the
following equation:

52. 4sinxcosx+2sinx+2cosx+1 = 0
Answer: Above equation can be written as
(4sinxcosx+2sinx) + (2cosx+1 ) = 0
⇨ 2sinx (2cosx+1) +
(2cosx+1) = 0
⇨ (2sinx+1) (2cosx+1)
= 0
⇨ (2sinx+1) = 0 or
2cosx+1 = 0
⇨ sinx = -1/2 or
cosx = -1/2
⇨ sinx = sin(π + )
or cosx = cos (π - )
⇨ x = nπ +(-1)
n

or x = 2nπ ± , ∀ n Є Z (Integers).

Question – 12 If cosx = - and π <x < , find the
value of sin3x and cos3x.
Answer: Since x lies in 3
rd
quadrant ∴ sinx is
negative.

53. Sinx = – = - then sin3x =
3sinx – sin
3
x = -
Cos3x = 4cos
3
x – 3cosx = -
[by putting the values of sinx & cosx]
Question – 13 (i) If cos(A+B) sin(C-D) = cos(A-B)
sin(C+D) , then show that
tanA tanB tanC + tanD = 0
(ii) If sinθ = n sin(θ+2), prove that
tan(θ+) = tan.
Answer: We can write above given result as
=
By C & D =
=
-cotA.cotB = tanC.cotD
- . = tanC.
⇨ - tanD = tanA tanB tanC ⇨ tanA tanB tanC + tanD = 0
.
(ii) = , by C & D =

54. ⇨ = ⇨ tan(θ+) = tan.
Question – 14 Prove that
(i) cos 52
0
+cos 68
0
+cos 172
0
= 0
(ii) sinA .sin(60
0
– A).sin(60
0
+ A) = sin3A.
(it lies in 2
nd
quad.)
Answer: (i) L.H.S. cos 52
0
+(cos 68
0
+cos 172
0
) =
cos52
0
+ 2 cos120
0
cos52
0

=
cos52
0
+ 2 ) cos52
0
[∵ cos120
0
= cos(180
0
-60
0
)]
= 0
(ii) L.H.S. sinA.[sin(60
0
– A).sin(60
0
+
A)] = sinA [sin
2
60
0
– sin
2
A]
=
sinA [ - sin
2
A] = [3sinA – 4sin
3
A]


= sin3A.
[We know that sin
2
A-sin
2
B = sin(A+B)sin(A-B)
& cos
2
A – sin
2
B = cos(A+B)cos(A-B)]

55. Question – 15 If sinx + siny = a and cosx + cosy = b, find
the values (i) (ii) .
Answer: sinx + siny = a ⇨ 2sin( cos) = a
..............1
cosx + cosy = b ⇨ 2cos( cos) = b
..............2
(i) By squaring & adding above results, we get
4 cos) [sin²( + cos²(] = a
2
+b
2

( 1 )
Sec
2
= ⇨ tan
2
= - 1 ⇨
tan = .
(ii) dividing 1 by 2, we get tan = .
Question – 16 Prove that
(i) tan70
0
= 2 tan50
0
+ tan20
0
.
(ii) Prove that: tan30
0
+ tan15
0
+ tan30
0
. tan15
0
=1.
(iii) cos
2
A + cos
2
B – 2cosAcosBcos(A+B) = sin
2
(A+B).
Solution: (i) We have tan70
0
= tan(50
0
+ 20
0
)
=

56. ⇨ tan70
0
[ ] =

⇨ tan70
0
- tan70
0
=

⇨ tan70
0
– tan(90
0
-20
0
) =

(cot20
0
= )
⇨ tan70
0
– tan50
0
=
⇨ tan70
0
= 2 tan50
0
+ tan20
0
.
(ii) [hint: take tan45
0
= tan(30
0
+15
0
)
(iii) L.H.S. cos
2
A + cos
2
B –
(2cosAcosB)cos(A+B) = cos
2
A + cos
2
B – (cos(A+B)+cos(A-
B))cos(A+B)
cos
2
A + cos
2
B – [cos
2
(A+B)+cos(A-B)cos(A+B)]
cos
2
A + cos
2
B – [cos
2
(A+B)+cos
2
(A) – sin
2
(B]
cos
2
A + cos
2
B - cos
2
(A+B)-cos
2
(A) + sin
2
(B]
1 - cos
2
(A+B) = sin
2
(A+B).
Question – 17 If x+y = , prove that (i)
(1+tanx)(1+tany) = 2 (ii) (cotx – 1)(coty – 1) = 2.
Answer: (i) tan(x+y) = tan ⇨ = 1
⇨ tanx+tany+tanx.tany = 1

57. ⇨ (1+ tanx) + tany(1 + tanx) = 1+1
⇨ (1+tanx)(1+tany) = 2
(ii) Similarly for second part by using cot(x+y) = .
Question – 18 Prove that (i) tan189
0
=
(ii) Find the value of tan22
0
30’.
Answer: (i) we can take both sides to prove above
result
L.H.S. tan189
0
= tan(180
0
+9
0
) = tan9° =
tan(45°-36°) =

= = .
(ii) Let x= 22
0
30’ then 2x = 45
0

We know that tanx = ⇨
tanx = = =
tan22
0
30’ = [∵ it lies in 1
st
quad.]
Question: If tanx+tany = a and cotx+coty = b, prove that
1/a - 1/b = cot(x+y).

58. Answer: L.H.S. = - =
{after simplification}.

Question – 19 Prove that cos
2
x + cos
2
(x+ ) + cos
2
(x-
) = .
Answer: L.H.S. cos
2
x + cos
2
(x+ ) +[1 - sin
2
(x- ) ]
⇨ 1+ cos
2
x + [ cos
2
(x+ ) - sin
2
(x- ) ]
⇨ 1+ cos
2
x + [ cos (x+ )
cos(x+-x+ ) ] ⇨ 1+ cos
2
x + [cos (2x) .cos() ]
( ∵cos(π - it lies in 2
nd
quad.)
⇨ 1+ cos
2
x + [cos (2x) )] ⇨
= = = .
Question – 20 Prove that
(a) sin
3
x + sin
3
(+x) + sin
3
( +x) = - sin3x.
[Hint: Use sin3A = 3sinA – 4sin
3
A ⇨ 4sin
3
A = 3sinA -
sin3A ⇨ sin
3
A = ¼[3sinA - sin3A]]
(b) If tanx + tan(x+) + tan+x) = 3 , then
show that tan3x = 1. [ use formula of tan(x+y)]

59. (c) Find in degrees and radians the angle
subtended b/w the hour hand and the minute hand
Of a clock at half past three. [answer is 75
0
,
5/12]
(d) - = cot 2A. [Hint
put cot3A = 1/tan3A & cotA = 1/tanA then take l.c.m.

and use formula tan(A-B).]
(e) If , are the distinct roots of acosθ +
bsinθ = c, prove that sin(+) = .
Solution of (e) If , are the distinct roots of acosθ +
bsinθ = c, then
acos + bsin = c & acos + bsin = c
By subtracting , we get a(cos – cos) + b(sin – sin) = 0
⇨ a(cos – cos) = b(sin – sin)
⇨ 2a sin sin = 2b cos sin
⇨ tan = ⇨ sin(+) = ⇨
sin(+) = .
Question: If sin2A = k sin2B, prove that =

60. [ Hint by c & d = , use formula of
sinx+siny= 2sin( cos)]
Question: If cos(x+2A) = n cosx, show that cotA =
tan(x+A).
Question: Prove that tan( = , if 2tan =
3tan.
[Hint: L.H.S. = , put the value of tan and
simplify it].
Question: If sinx + siny = a and cosx + cosy = b,find the value
of cos(x-y).
Answer: squaring and adding above results, we will get
cos(x-y) = ½[a
2
+b
2
– 2]
Question: Show that - =4
[Hint: 2 [ - ] = 2 [ - ]
Question: Prove that : = tan(x/2)
[ Hint: use 1 – cosx = 2sin
2
x/2 , 1+cosx = 2cos
2
x/2 and sinx
= 2sinx/2.cosx/2]
** Question Solve: sec∅ + tan∅ = 1

61. Solution + sin∅ = cos∅ ⇨ cos∅ - sin∅ =
dividing by = ∵ a=1,b=1
(cos∅ - sin∅ )/ = 1 ⇨ cos(п/4) cos∅ -
sin(п/4) sin∅ =1 ⇨ cos(∅+п/4) = cos0
0

∅+п/4 = 2nп±0, n∈Z ⇨ ∅ = 2nп – п/4.
** Question: If tan
2
A = 2tan
2
B + 1, prove that cos2A +
sin
2
B =0.
Answer: L.H.S. + sin
2
B = + sin
2
B ,
by putting above result and simplify it.
** Question: Find the maximum and minimum values of
sinx+cosx.
Answer: maximum value of (asinx+bcosx) = =

Minimum value of (asinx+bcosx) = - =
-
Or
[ sinx + cosx] = sin(x+п/4), as -1 ≤
sin(x+п/4) ≤1 ∀ x
**Question: Find the minimum value of 3cosx+4sinx+8. [
answer is 3 as above result]

62. **Question: ∀ x in (0,п/2), show that cos(sinx) >
sin(cosx).
Answer: п/2> [∵п/2 = 22/7=1.57 and =1.4]
We know that п/2 > ≥ sinx+cosx ⇨ п/2 -
sinx > cosx ⇨ cos(sinx) > sin(cosx).
** Question: If A = cos
2
x + sin
4
x ∀ x, prove that ¾ ≤ A ≤
1
Answer: A = cos
2
x + sin
2
x. sin
2
x ≤ cos
2
x + sin
2
x ⇨ A ≤ 1 ,
A = (1 - sin
2
x) + sin
4
x = [sin
2
x – (½)]
2
+ (¾) ≥ ¾.
** Question: (i) find the greatest value of sinx.cosx [
½.(2sinx.cosx) ≤ ½]
(ii) If sinx and cosx are the roots of ax
2
– bx+c
= 0, then a
2
– b
2
+2ac = 0.
[ Hint: sum and product of roots ⇨ (sinx+cosx)
2
=
1+2(c/a)]
** Question : If f(x) = cos
2
x+sec
2
x , then find which is
correct f(x)<1, f(x)=1, 2< f(x)<1, f(x)≥2.
[A.M. ≥ G.M.⇨ f(x)/2= (cos
2
x+sec
2
x)/2 ≥ (cos
2
x.sec
2
x) ,
ANSWER IS f(x)≥2].
** Question: solve = 28.
[hint: use 2cos
2
x = 1+ cos2x , 2sin
2
x = 1 – cos2x and let
sin2x+cos2x = y

63. Above equation becomes = 28. Then
put t = ⇨ 3t
2
– 28t +9=0 ⇨ t=1/3 , 9
When y=-1 then sin2x+cos2x = -1 ⇨ cosx = 0 or tanx = -1
When y = 2 then sin2x+cos2x =2 ⇨ sin2x+ cos2x =
⇨ sin(2x+ >1 which is not possible