Class 11th Maths – RD Sharma Chapter 9 : Value of Trigonometric Functions at Multiples and Submultiples of an angle – Exercise 9.1 Solutions (Step-by-Step Guide)

    Prove the following identities (1 – 27) :

1. √((1 − cos 2x)/(1 + cos 2x)) = tan x Watch Solution

2. sin 2x/(1 − cos 2x) = cot x Watch Solution

3. sin 2x/(1 + cos 2x) = tan x Watch Solution

4. √(2 + √(2 + 2 cos 4x)) = 2 cos x, 0 < x < π/4 Watch Solution

5. (1 − cos 2x + sin 2x)/(1 + cos 2x + sin 2x) = tan x Watch Solution

6. (sin x + sin 2x)/(1 + cos x + cos 2x) = tan x Watch Solution

7. cos 2x/(1 + sin 2x) = tan (π/4 − x) Watch Solution

8. cos x/(1 − sin x) = tan (π/4 + x/2) Watch Solution

9. cos² π/8 + cos² 3π/8 + cos² 5π/8 + cos² 7π/8 = 2 Watch Solution

10. sin² π/8 + sin² 3π/8 + sin² 5π/8 + sin² 7π/8 = 2 Watch Solution

11. (cos α + cos β)² + (sin α + sin β)² = 4 cos² ((α − β)/2) Watch Solution

12. sin² (π/8 + x/2) − sin² (π/8 − x/2) = (1/√2) sin x Watch Solution

13. 1 + cos² 2x = 2 (cos⁴ x + sin⁴ x) Watch Solution

14. cos³ 2x + 3 cos 2x = 4 (cos⁶ x − sin⁶ x) Watch Solution

15. (sin 3x + sin x) sin x + (cos 3x − cos x) cos x = 0 Watch Solution

16. cos² (π/4 − x) − sin² (π/4 − x) = sin 2x Watch Solution

17. cos 4x = 1 − 8 cos² x + 8 cos⁴ x Watch Solution

18. sin 4x = 4 sin x cos³ x − 4 cos x sin³ x Watch Solution

19. 3 (sin x − cos x)⁴ + 6 (sin x + cos x)² + 4 (sin⁶ x + cos⁶ x) = 13 Watch Solution

20. 2 (sin⁶ x + cos⁶ x) − 3 (sin⁴ x + cos⁴ x) + 1 = 0 Watch Solution

21. cos⁶ x − sin⁶ x = cos 2x (1 − (1/4) sin² 2x) Watch Solution

22. tan (π/4 + x) + tan (π/4 − x) = 2 sec 2x Watch Solution

23. cot² x − tan² x = 4 cot 2x cosec 2x Watch Solution

24. cos 4x − cos 4α = 8 (cos x − cos α)(cos x + cos α)(cos x − sin α)(cos x + sin α) Watch Solution

25. sin 3x + sin 2x − sin x = 4 sin x cos x/2 cos 3x/2 Watch Solution

26. Prove that : tan 82 1/2° = (√3 + √2)(√2 + 1) = √2 + √3 + √4 + √6 Watch Solution

27. Prove that : cot π/8 = √2 + 1 Watch Solution

28. (i) If cos x = −3/5 and x lies in the IIIrd quadrant, find the values of cos x/2, sin x/2 and sin 2x. Watch Solution

            (ii) If cos x = −3/5 and x lies in IInd quadrant, find the values of sin 2x and sin x/2. Watch Solution

29. If sin x = √5/3 and x lies in IInd quadrant, find the values of cos x/2, sin x/2 and tan x/2 Watch Solution

30. (i) If 0 ≤ x ≤ π and x lies in the IInd quadrant such that sin x = 1/4. Find the values of cos x/2, sin x/2 and tan x/2. Watch Solution

           (ii) If cos x = 4/5 and x is acute, find tan 2x Watch Solution

          (iii) If sin x = 4/5 and 0 < x < π/2 , find the value of sin 4x. Watch Solution

31. If tan x = b/a , then find the value of √((a+b)/(a−b)) + √((a−b)/(a+b)). Watch Solution

32. If tan A = 1/7 and tan B = 1/3 , show that cos 2A = sin 4B. Watch Solution

33. Prove that: cos 7° cos 14° cos 28° cos 56° = sin 68°/(16 cos 83°) Watch Solution

34. Prove that: cos 2π/15 cos 4π/15 cos 8π/15 cos 16π/15 = 1/16 Watch Solution

35. Prove that: cos π/5 cos 2π/5 cos 4π/5 cos 8π/5 = 1/16 Watch Solution

36. Prove that: cos π/65 cos 2π/65 cos 4π/65 cos 8π/65 cos 16π/65 cos 32π/65 = 1/64  Watch Solution

37. If 2 tan α = 3 tan β, prove that tan (α − β) = sin 2β/(5 − cos 2β). Watch Solution

38. If sin α + sin β = a and cos α + cos β = b, prove that

        (i) sin (α + β) = 2ab/(a² + b²) Watch Solution

       (ii) cos (α − β) = (a² + b² − 2)/2 Watch Solution

39. If 2 tan α/2 = tan β/2 , prove that cos α = (3 + 5 cos β)/(5 + 3 cos β). Watch Solution

40. If cos x = (cos α + cos β)/(1 + cos α cos β) , prove that tan x/2 = ± tan α/2 tan β/2 Watch Solution

41. If sec (x + α) + sec (x − α) = 2 sec x, prove that cos x = ± √2 cos α/2 Watch Solution

42. If cos α + cos β = 1/3 and sin α + sin β = 1/4 , prove that cos ((α − β)/2) = ± 5/24. Watch Solution

43. If sin α = 4/5 and cos β = 5/13 , prove that cos ((α − β)/2) = 8/√65. Watch Solution

44. If a cos 2x + b sin 2x = c has α and β as its roots, then prove that

           (i) tan α + tan β = 2b/(a + c)

          (ii) tan α tan β = (c − a)/(c + a)

          (iii) tan (α + β) = b/a Watch Solution

45. If cos α + cos β = 0 = sin α + sin β, then prove that cos 2α + cos 2β = −2 cos (α + β). Watch Solution