Loading [MathJax]/jax/element/mml/optable/GreekAndCoptic.js

GSEB Solutions Class 11 Maths Chapter 3 Trigonometric Functions Ex 3.3

Gujarat Board GSEB Textbook Solutions Class 11 Maths Chapter 3 Trigonometric Functions Ex 3.3 Textbook Questions and Answers.

Gujarat Board Textbook Solutions Class 11 Maths Chapter 3 Trigonometric Functions Ex 3.3

Prove that:
1. sin2 \frac{π}{6} + cos2 \frac{π}{3} – tan2 \frac{π}{4} = – \frac{1}{2}
2. 2 sin2 \frac{π}{6} + cosec2 \frac{7π}{6} cos2 \frac{π}{3} = \frac{3}{2}
3. cot2 \frac{π}{6} + cosec \frac{5π}{6} + 3 tan2 \frac{π}{6} = 6
4. 2 sin2 \frac{3π}{4} + 2 cos2 \frac{π}{4} + 2 sec2 \frac{π}{3} = 10
Solutions to questions 1 – 4:
1. L.H.S. = sin2 \frac{π}{6} + cos2 \frac{π}{3} – tan2 \frac{π}{4}
= (sin \frac{π}{6})2 + (cos \frac{π}{3})2 – (tan \frac{π}{4})2
= (\frac{1}{2})2 + (\frac{1}{2})2 – 12
[∵ sin \frac{π}{6} = \frac{1}{2}, cos \frac{π}{3} = \frac{1}{2}, tan \frac{π}{4} = 1]
= \frac{1}{4} + \frac{1}{4} – 1 = \frac{1}{2} – 1
= \frac{- 1}{2} = R.H.S.

2. L.H.S. = 2 sin2 \frac{π}{6} + cosec2 \frac{7π}{6} cos2 \frac{π}{3}
= 2(sin \frac{π}{6})2 + (cosec \frac{7π}{6})2 (cos \frac{π}{3})2
= 2(\frac{1}{2})2 + [cosec (π + \frac{π}{6})]2 (\frac{1}{2})2
= 2 × \frac{1}{4} + (- cosec \frac{π}{6})2 ((\frac{1}{4}) [∵ cosec (π + θ) = – cosec θ]
= \frac{1}{2} + (2)2 \frac{1}{4} = \frac{1}{2} + 1 = \frac{3}{2} = R.H.S.

3. L.H.S. = cot2 \frac{π}{6} + cosec \frac{5π}{6} + 3 tan2 \frac{π}{6}
= (cot \frac{π}{6})2 + cosec (π – \frac{π}{6}) + 3(tan \frac{π}{6})2
= (\sqrt{3})2 + cosec \frac{π}{6} + 3(\frac{1}{\sqrt{3}})2
= 3 + 2 + 3 × \frac{1}{3} = 3 + 2 + 1 = 6 = R.H.S.

4. L.H.S. = 2 sin2 \frac{3π}{4} + 2 cos2 \frac{π}{4} + 2 sec2 \frac{π}{3}
GSEB Solutions Class 11 Maths Chapter 3 Trigonometric Functions Ex 3.3 img 1
= 1 + 1 + 8 = 10 = R.H.S.

5. Find the value of:
(i) sin 75°
(ii) tan 15°
Solution:
(i) sin 75° = sin (45° + 30°)
= sin 45° cos 30° + cos 45° sin 30° [sin (A + B) = sin A cos B + cos A sin B]
GSEB Solutions Class 11 Maths Chapter 3 Trigonometric Functions Ex 3.3 img 2

(ii) tan 15° = tan (45° – 30°)
GSEB Solutions Class 11 Maths Chapter 3 Trigonometric Functions Ex 3.3 img 3

GSEB Solutions Class 11 Maths Chapter 3 Trigonometric Functions Ex 3.3

Prove the following:
6. cos (\frac{π}{4} – x) cos (\frac{π}{4} – y) – sin (\frac{π}{4} – x) sin (\frac{π}{4} – y) = sin (x + y)
7. \frac{\tan \left(\frac{\pi}{4}+x\right)}{\tan \left(\frac{\pi}{4}-x\right)}=\frac{(1+\tan x)^{2}}{(1-\tan x)^{2}}
8. \frac{\cos (\pi+x) \cos (-x)}{\sin (\pi-x) \cos \left(\frac{\pi}{2}+x\right)} = cot2 x
9. cos (\frac{3π}{2} + x)cos (2π + x) [cos ((\frac{3π}{2} – x) + cot (2π + x)] = 1
Solutions to questions 6 – 9:

6. L.H.S. = cos (\frac{π}{4} – x)cos(\frac{π}{4} – y) – sin (\frac{π}{4} – x)sin(\frac{π}{4} – y)
Put \frac{π}{4} – x = A and \frac{π}{4} – y = B, we get:
L.H.S. = cos A cos B – sin A sin B
= cos (A + B)
= cos[(\frac{π}{4} – x) + (\frac{π}{4} – y)] = cos [\frac{π}{2} – (x + y)] [∵cos (\frac{π}{2} – θ) = sin θ]
= sin (x + y) = R.H.S.

7.
GSEB Solutions Class 11 Maths Chapter 3 Trigonometric Functions Ex 3.3 img 14

8. L.H.S. = \frac{\cos (\pi+x) \cos (-x)}{\sin (\pi-x) \cos \left(\frac{\pi}{2}+x\right)}
Now, cos (π + x) = – cos x, cos (- x) = cos x,
sin (π – x) = sin x, cos (\frac{π}{2} + x) = – sin x
GSEB Solutions Class 11 Maths Chapter 3 Trigonometric Functions Ex 3.3 img 4
= cot2x = R.H.S.

9. L.H.S. = cos (\frac{3π}{2} + x) cos (2π + x) [cot (\frac{3π}{2} – x) + cot (2π + x)]
Now, cos (\frac{3π}{2} + x) = sin x, cos (2π + x) = cos x,
cot (\frac{3π}{2} – x) = tan x, cot (2π + x) = cot x.
∴ L.H.S. = sin x cos x [(tan x) + cot x]
GSEB Solutions Class 11 Maths Chapter 3 Trigonometric Functions Ex 3.3 img 5
= 1 = R.H.S.

GSEB Solutions Class 11 Maths Chapter 3 Trigonometric Functions Ex 3.3

Prove the following:
10. sin (n + 1)x sin (n + 2)x + cos (n + 1)x cos (n + 2)x = cos x
11. cos (\frac{3π}{4} + x) – cos (\frac{3π}{4} – x) = – \sqrt{2} sin x
12. sin26x – sin24x = sin 2x sin 8x
13. cos22x – cos26x = sin 4x sin 8x
14. sin 2x + 2 sin 4x + sin 6x = 4 cos2x sin 4x
15. cot 4x(sin 5x + sin 3x) = cot x(sin 5x – sin 3x)
Solutions to questions 10 – 15:
10. L.H.S. = sin (n + 1)x sin (n + 2)x + cos (n + 1)x cos (n + 2)x
= cos (n + 2)x cos (n + 1)x + sin (n + 2)x sin (n + 1)x
Put (n + 2)x = A, (n + 1)x = B.
L.H.S. = cos A cos B + sin A sin B
= cos (A – B)
= cos [(n + 2)x – (n + 1)x]
= cos x = R.H.S.

11. L.H.S. = cos(\frac{3π}{4} + x) – cos(\frac{3π}{4} – x)
Put \frac{3π}{4} + x = θ, \frac{3π}{4} – x = ϕ.
∴ L.H.S. = cos θ – cos ϕ
GSEB Solutions Class 11 Maths Chapter 3 Trigonometric Functions Ex 3.3 img 6

12. L.H.S. = sin2 6x – sin2 4x
We apply the formula sin2A – sin2B = sin (A + B) sin (A – B).
= (sin A cos B + cos A sin B)(sin A cos B – cos A sin B)
= sin2A cos2B – cos2A sin2B
= sin2A (1 – sin2B) – (1 – sin2 A) sin2B
= sin2A – sin2A sin2B – sin2 B + sin2A sin2B
= sin2A – sin2B
∴ L.H.S. = sin26x – sin2 4x = sin (6x + 4x) sin (6x – 4x)
= sin 10x sin 2x = R.H.S.

13. L.H.S. = cos22x – cos2 6x = 1 – sin2 2x – (1 – sin2 6x)
= sin2 6x – sin2 2x = sin (6x + 2x)sin (6x – 2x)
= sin 8x sin 4x = R.H.S.

14. L.H.S. = sin 2x + 2sin 4x + sin 6x
= (sin 6x + sin 2x) + 2 sin 4x
= 2 sin \frac{6x+2x}{2} cos \frac{6x-2x}{2} + 2sin 4x
[∵ sin C + sin D = 2 sin \frac{C+D}{2} cos \frac{C-D}{2}]
= 2sin 4x cos 2x + 2sin 4x
= 2sin 4x [cos 2x + 1]
= 2 sin 4x 2cos2x
[∵ cos 2x + 1 = 2cos2x – 1 + 1 = 2cos2 x]
= 2sin 4x 2cos2x
= 4sin 4x cos2x = R.H.S.

15. L.H.S. = cot 4x [sin 5x + sin 3x]
= cot 4x [2sin 4x cosx]
= \frac{cos x}{sin x} × 2sin 4x cos x
= 2cosx cos 4x
= \frac{cos x}{sin x} × (2cos 4x sin x)
= \frac{cos x}{sin x} × (sin 5x – sin 3x) = cot x (sin 5x – sin 3x)

GSEB Solutions Class 11 Maths Chapter 3 Trigonometric Functions Ex 3.3

Prove the following:
16. \frac{cos 9x – cos 5x}{sin 17x – sin 3x} = – \frac{sin x}{cos 10x}
17. \frac{sin 5x + sin 3x}{cos 5x + cos 3x} = tan 4x
18. \frac{sin x – sin y}{cos x + cos y} = tan \frac{x – y}{2}
19. \frac{sin x + sin 3x}{cos x + cos 3x} = tan 2x
20. \frac{\sin x-\sin 3 x}{\sin ^{2} x-\cos ^{2} x} = 2 sin x
Solutions to questions 16 – 20:
16. L.H.S. =
GSEB Solutions Class 11 Maths Chapter 3 Trigonometric Functions Ex 3.3 img 7

17. L.H.S. =
GSEB Solutions Class 11 Maths Chapter 3 Trigonometric Functions Ex 3.3 img 8
= \frac{sin 4x}{cos 4x} = tan 4x = R.H.S.

18. L.H.S. =
GSEB Solutions Class 11 Maths Chapter 3 Trigonometric Functions Ex 3.3 img 9

19. L.H.S. =
GSEB Solutions Class 11 Maths Chapter 3 Trigonometric Functions Ex 3.3 img 10

20. L.H.S.=
GSEB Solutions Class 11 Maths Chapter 3 Trigonometric Functions Ex 3.3 img 11

GSEB Solutions Class 11 Maths Chapter 3 Trigonometric Functions Ex 3.3

Prove the following:
21. \frac{cos 4x + cos 3x + cos 2x}{sin 4x + sin 3x + sin 2x} = cot 3x
22. cot x cot 2x – cot 2x cot 3x – cot 3x cot x = 1
23. tan 4x = \frac{4 \tan x\left(1-\tan ^{2} x\right)}{1-6 \tan ^{2} x+\tan ^{4} x}
24. cos 4x = 1 – 8 sin2x cos2 x
25. cos 6x = 32 cos6x = 48cos4x + 18cos2x – 1
Solutions to questions 21 – 25:
21. L.H.S. =
GSEB Solutions Class 11 Maths Chapter 3 Trigonometric Functions Ex 3.3 img 12

22. L.H.S. = cot x cot 2x – cot 2x cot 3x – cot 3x cot x
We have: 3x = x + 2x
cot 3x = cot (x + 2x)
= \frac{cot xcot2x – 1}{cot x + cot 2x}
By cross multiplication,
cot 3x (cot x + cot 2x) = cot x cot 2x – 1
cot x cot 3x + cot 2x cot 3x = cos x cot 2x – 1
∴ cos x cot 2x – cot 2x cot 3x – cot 3x cot x = 1 = R.H.S.

23. L.H.S. = tan 4x = tan 2(2x)
GSEB Solutions Class 11 Maths Chapter 3 Trigonometric Functions Ex 3.3 img 13

24. L.H.S. = cos 4x = cos 2(2x)
= 2cos2 2x – 1 [∵ cos 2A = 2cos2A – 1]
= 2[2 cos2 x – 1]2 – 1
= 2[4 cos4x – 4 cos2x + 1] – 1
= 8cos4x – 8cos2x + 2 – 1
= 1 – 8 cos2x (1 – cos2x) = 1 – 8cos2xsin2x = R.H.S.

25. cos 6x = cos 3(2x) = 4 cos32x – 3cos 2x
Putting cos 2x = 2cos2x – 1, we get [∵ cos 3A = cos (2A + A) = cos 2AcosA – sin 2Asin A]
= (2cos2A – 1)cos A – 2cos Asin2 A
= 2cos3A – cos A – 2cos A(1 – cos2 A)
= 4 cos3A – 3 cos A
∴ cos 6x = 4(2cos2x – 1)3 – 3(2cos2 x – 1)
= 4(8 cos6x – 12cos4x + 6cos2 x – 1) – 6cos2x + 3
= 32cos6x – 48cos4x + 18cos2x – 1 = R.H.S.

Leave a Comment

Your email address will not be published. Required fields are marked *