Class 11th Maths – RD Sharma Chapter 8 : Trigonometric Formulae – Exercise 8.2 Solutions (Step-by-Step Guide)

1. Express each of the following as the product of sines and cosines:

  (i) sin 12x + sin 4x Watch Solution

  (ii) sin 5x − sin x Watch Solution

  (iii) cos 12x + cos 8x Watch Solution

  (iv) cos 12x − cos 4x Watch Solution

  (v) sin 2x + cos 4x Watch Solution

2. Prove that:

  (i) sin 38° + sin 22° = sin 82° Watch Solution

  (ii) cos 100° + cos 20° = cos 40° Watch Solution

  (iii) sin 50° + sin 10° = cos 20° Watch Solution

  (iv) sin 23° + sin 37° = cos 7° Watch Solution

  (v) sin 105° + cos 105° = cos 45° Watch Solution

  (vi) sin 40° + sin 20° = cos 10° Watch Solution

3. Prove that:

  (i) cos 55° + cos 65° + cos 175° = 0 Watch Solution

  (ii) sin 50° − sin 70° + sin 10° = 0 Watch Solution

  (iii) cos 80° + cos 40° − cos 20° = 0 Watch Solution

  (iv) cos 20° + cos 100° + cos 140° = 0 Watch Solution

  (v) sin 5π/18 − cos 4π/9 = √3 sin π/9 Watch Solution

  (vi) cos π/12 − sin π/12 = 1/√2 Watch Solution

  (vii) sin 80° − cos 70° = cos 50° Watch Solution

  (viii) sin 51° + cos 81° = cos 21° Watch Solution

4. Prove that:

  (i) cos (3π/4 + x) − cos (3π/4 − x) = −√2 sin x Watch Solution

  (ii) cos (π/4 + x) + cos (π/4 − x) = √2 cos x Watch Solution

5. Prove that:

  (i) sin 65° + cos 65° = √2 cos 20° Watch Solution

   (ii) sin 47° + cos 77° = cos 17° Watch Solution

6. Prove that:

  (i) cos 3A + cos 5A + cos 7A + cos 15A = 4 cos 4A cos 5A cos 6A Watch Solution

  (ii) cos A + cos 3A + cos 5A + cos 7A = 4 cos A cos 2A cos 4A Watch Solution

  (iii) sin A + sin 2A + sin 4A + sin 5A = 4 cos A/2 cos 3A/2 sin 3A Watch Solution

  (iv) sin 3A + sin 2A − sin A = 4 sin A cos A/2 cos 3A/2 Watch Solution

   (v) cos 20° cos 100° + cos 100° cos 140° − cos 140° cos 200° = −3/4 Watch Solution

  (vi) sin x/2 sin 7x/2 + sin 3x/2 sin 11x/2 = sin 2x sin 5x Watch Solution

  (vii) cos x cos x/2 − cos 3x cos 9x/2 = sin 7x sin 8x Watch Solution

7. Prove that:

  (i) (sin A + sin 3A)/(cos A − cos 3A) = cot A Watch Solution

  (ii) (sin 9A − sin 7A)/(cos 7A − cos 9A) = cot 8A Watch Solution

  (iii) (sin A − sin B)/(cos A + cos B) = tan (A − B)/2 Watch Solution

  (iv) (sin A + sin B)/(sin A − sin B) = tan (A + B)/2 cot (A − B)/2 Watch Solution

  (v) (cos A + cos B)/(cos B − cos A) = cot (A + B)/2 cot (A − B)/2 Watch Solution

8. Prove that:

  (i) (sin A + sin 3A + sin 5A)/(cos A + cos 3A + cos 5A) = tan 3A Watch Solution

  (ii) (cos 3A + 2 cos 5A + cos 7A)/(cos A + 2 cos 3A + cos 5A) = (cos 5A)/(cos 3A) Watch Solution

  (iii) (cos 4A + cos 3A + cos 2A)/(sin 4A + sin 3A + sin 2A) = cot 3A Watch Solution

  (iv) (sin 3A + sin 5A + sin 7A + sin 9A)/(cos 3A + cos 5A + cos 7A + cos 9A) = tan 6A Watch Solution

  (v) (sin 5A − sin 7A + sin 8A − sin 4A)/(cos 4A + cos 7A − cos 5A − cos 8A) = cot 6A Watch Solution

  (vi) (sin 5A cos 2A − sin 6A cos A)/(sin A sin 2A − cos 2A cos 3A) = tan A Watch Solution

  (vii) (sin 11A sin A + sin 7A sin 3A)/(cos 11A sin A + cos 7A sin 3A) = tan 8A Watch Solution

  (viii) (sin 3A cos 4A − sin A cos 2A)/(sin 4A sin A + cos 6A cos A) = tan 2A Watch Solution

  (ix) (sin A sin 2A + sin 3A sin 6A)/(sin A cos 2A + sin 3A cos 6A) = tan 5A Watch Solution

  (x) (sin A + 2 sin 3A + sin 5A)/(sin 3A + 2 sin 5A + sin 7A) = (sin 3A)/(sin 5A) Watch Solution

  (xi) (sin (θ + ϕ) − 2 sin θ + sin (θ − ϕ))/(cos (θ + ϕ) − 2 cos θ + cos (θ − ϕ)) = tan θ Watch Solution

9. Prove that:

  (i) sin α + sin β + sin γ − sin (α + β + γ) = 4 sin (α + β)/2 sin (β + γ)/2 sin (γ + α)/2 Watch Solution

  (ii) cos (A + B + C) + cos (A − B + C) + cos (A + B − C) + cos (−A + B + C) = 4 cos A cos B cos C Watch Solution

10. If cos A + cos B = 1/2 and sin A + sin B = 1/4 , prove that: tan (A + B)/2 = 1/2 Watch Solution

11. If cosec A + sec A = cosec B + sec B, prove that: tan A tan B = cot (A + B)/2 Watch Solution

12. If sin 2A = λ sin 2B, prove that: tan (A + B)/tan (A − B) = (λ + 1)/(λ − 1) Watch Solution

13. Prove that:

  (i) [cos (A + B + C) + cos (−A + B + C) + cos (A − B + C) + cos (A + B − C)] / [sin (A + B + C) + sin (−A + B + C) + sin (A − B + C) − sin (A + B − C)] = cot C Watch Solution

  (ii) sin (B − C) cos (A − D) + sin (C − A) cos (B − D) + sin (A − B) cos (C − D) = 0 Watch Solution

14. If cos (A − B)/cos (A + B) + cos (C + D)/cos (C − D) = 0, prove that tan A tan B tan C tan D = −1 Watch Solution

15. If cos (α + β) sin (γ + δ) = cos (α − β) sin (γ − δ), prove that cot α cot β cot γ = cot δ Watch Solution

16. If y sin ϕ = x sin (2θ + ϕ), prove that (x + y) cot (θ + ϕ) = (y − x) cot θ Watch Solution

17. If cos (A + B) sin (C − D) = cos (A − B) sin (C + D), prove that tan A tan B tan C + tan D = 0 Watch Solution

18. If x cos θ = y cos (θ + 2π/3) = z cos (θ + 4π/3), prove that xy + yz + zx = 0 Watch Solution

19. If m sin θ = n sin (θ + 2α), prove that tan (θ + α) cot α = (m + n)/(m − n). Watch Solution