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

Chapter 9: Values of Trigonometric Functions at Multiples and Submultiples of an Angle – R. D. Sharma Class 11th Maths

1. Values of Trigonometric Functions at Multiples and Submultiples of an Angle Exercise 9.1 Video Solution

2. Values of Trigonometric Functions at Multiples and Submultiples of an Angle Exercise 9.2 Video Solution

3. Values of Trigonometric Functions at Multiples and Submultiples of an Angle Exercise 9.3 Video Solution

4. Values of Trigonometric Functions at Multiples and Submultiples of an Angle Multiple Choice Questions (MCQs) Video Solution Video Solution

5. Values of Trigonometric Functions at Multiples and Submultiples of an Angle Fill in the Blanks (FBQs) Video Solution

6. Values of Trigonometric Functions at Multiples and Submultiples of an Angle Very Short Answer Questions (VSAQs) Video Solution Video Solution