Class 11th Maths – RD Sharma Chapter 7 : Value of Trigonometric Functions at Sum or Difference of Angles – Exercise 7.1 Solutions

1. If sin A = 4/5 and cos B = 5/13 , where 0 < A, B < π/2 , find the values of the following:

  (i) sin (A + B)
  (ii) cos (A + B)
  (iii) sin (A − B)
  (iv) cos (A − B) Watch Solution

2. (a) If sin A = 12/13 and sin B = 4/5 , where π/2 < A < π and 0 < B < π/2 , find the following:

  (i) sin (A + B)
  (ii) cos (A + B) Watch Solution

  (b) If sin A = 3/5 , cos B = −12/13 , where A and B both lie in second quadrant, find the value of sin (A + B).

3. If cos A = −24/25 and cos B = 3/5 , where π < A < 3π/2 and 3π/2 < B < 2π , find the following:

  (i) sin (A + B)
  (ii) cos (A + B) Watch Solution

4. If tan A = 3/4 , cos B = 9/41 , where π < A < 3π/2 and 0 < B < π/2 , find tan (A + B). Watch Solution
5. If sin A = 1/2 , cos B = 12/13 , where π/2 < A < π and 3π/2 < B < 2π , find tan (A − B). Watch Solution
6. If sin A = 1/2 , cos B = √3/2 , where π/2 < A < π and 0 < B < π/2 , find the following: 

  (i) tan (A + B)
  (ii) tan (A − B) Watch Solution

7. Evaluate the following:

   (i) sin 78° cos 18° − cos 78° sin 18° Watch Solution

  (ii) cos 47° cos 13° − sin 47° sin 13° Watch Solution

  (iii) sin 36° cos 9° + cos 36° sin 9° Watch Solution

  (iv) cos 80° cos 20° + sin 80° sin 20° Watch Solution

8. If cos A = −12/13 and cot B = 24/7 , where A lies in the second quadrant and B in the third quadrant, find the values of the following:

  (i) sin (A + B)
  (ii) cos (A + B)
  (iii) tan (A + B) Watch Solution

9. Prove that:
   cos 7π/12 + cos π/12 = sin 5π/12 − sin π/12 Watch Solution
10. Prove that:
    (tan A + tan B)/(tan A − tan B) = sin (A + B)/sin (A − B) Watch Solution
11. Prove that:

  (i) (cos 11° + sin 11°)/(cos 11° − sin 11°) = tan 56° Watch Solution

  (ii) (cos 9° + sin 9°)/(cos 9° − sin 9°) = tan 54° Watch Solution

  (iii) (cos 8° − sin 8°)/(cos 8° + sin 8°) = tan 37° Watch Solution

12. Prove that:

  (i) sin (π/3 − x) cos (π/6 + x) + cos (π/3 − x) sin (π/6 + x) = 1 Watch Solution

  (ii) sin (4π/9 + 7) cos (π/9 + 7) − cos (4π/9 + 7) sin (π/9 + 7) = √3/2 Watch Solution

  (iii) sin (3π/8 − 5) cos (π/8 + 5) + cos (3π/8 − 5) sin (π/8 + 5) = 1 Watch Solution

13. Prove that:
    (tan 69° + tan 66°)/(1 − tan 69° tan 66°) = −1 Watch Solution
14. (i) If tan A = 5/6 and tan B = 1/11 , prove that A + B = π/4 Watch Solution

           (ii) If tan A = m/(m − 1) and tan B = 1/(2m − 1) , then prove that A − B = π/4 Watch Solution

15. Prove that:

  (i) cos² π/4 − sin² π/12 = √3/4 Watch Solution

  (ii) sin² (n + 1)A − sin² nA = sin (2n + 1)A sin A Watch Solution

16. Prove that:

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

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

  (iii) sin (A − B)/(sin A sin B) + sin (B − C)/(sin B sin C) + sin (C − A)/(sin C sin A) = 0 Watch Solution

  (iv) sin² B = sin² A + sin² (A − B) − 2 sin A cos B sin (A − B) Watch Solution

  (v) cos² A + cos² B − 2 cos A cos B cos (A + B) = sin² (A + B) Watch Solution

  (vi) tan (A + B)/cot (A − B) = (tan² A − tan² B)/(1 − tan² A tan² B) Watch Solution

17. Prove that:

  (i) tan 8x − tan 6x − tan 2x = tan 8x tan 6x tan 2x Watch Solution

  (ii) tan π/12 + tan π/6 + tan π/12 tan π/6 = 1 Watch Solution

  (iii) tan 36° + tan 9° + tan 36° tan 9° = 1 Watch Solution

  (iv) tan 13x − tan 9x − tan 4x = tan 13x tan 9x tan 4x Watch Solution

18. Prove that:
    (tan² 2x − tan² x)/(1 − tan² 2x tan² x) = tan 3x tan x Watch Solution
19. If sin (x + y)/sin (x − y) = (a + b)/(a − b) , show that tan x/tan y = a/b. Watch Solution
20. If tan A = x tan B, prove that sin (A − B)/sin (A + B) = (x − 1)/(x + 1). Watch Solution
21. If tan (A + B) = x and tan (A − B) = y, find the values of tan 2A and tan 2B. Watch Solution
22. If cos A + sin B = m and sin A + cos B = n, prove that 2 sin (A + B) = m² + n² − 2. Watch Solution
23. If tan A + tan B = a and cot A + cot B = b, prove that: cot (A + B) = 1/a − 1/b. Watch Solution
24. If x lies in the first quadrant and cos x = 8/17 , then prove that:
    cos (π/6 + x) + cos (π/4 − x) + cos (2π/3 − x) = ((√3 − 1)/2 + 1/√2) 23/17 Watch Solution
25. If tan x + tan (x + π/3) + tan (x + 2π/3) = 3, then prove that
    (3 tan x − tan³ x)/(1 − 3 tan² x) = 1. Watch Solution
26. If sin (α + β) = 1 and sin (α − β) = 1/2 , where 0 ≤ α, β ≤ π/2 , then find the values of tan (α + 2β) and tan (2α + β). Watch Solution
27. If α, β are two different values of x lying between 0 and 2π which satisfy the equation 6 cos x + 8 sin x = 9, find the value of sin (α + β). Watch Solution
28. If sin α + sin β = a and cos α + cos β = b, show that

  (i) sin (α + β) = 2ab/(a² + b²)

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

29. Prove that:

  (i) 1/[sin (x − a) sin (x − b)] = [cot (x − a) − cot (x − b)]/sin (a − b) Watch Solution

  (ii) 1/[sin (x − a) cos (x − b)] = [cot (x − a) + tan (x − b)]/cos (a − b) Watch Solution

  (iii) 1/[cos (x − a) cos (x − b)] = [tan (x − b) − tan (x − a)]/sin (a − b) Watch Solution

30. If sin α sin β − cos α cos β + 1 = 0, prove that 1 + cot α tan β = 0. Watch Solution
31. If tan α = x + 1, tan β = x − 1, show that 2 cot (α − β) = x². Watch Solution
32. If angle θ is divided into two parts such that the tangents of one part is λ times the tangent of other, and ϕ is their difference, then show that
    sin θ = (λ + 1)/(λ − 1) sin ϕ. Watch Solution
33. If tan x = (sin α − cos α)/(sin α + cos α) , then show that sin α + cos α = √2 cos x. Watch Solution
34. If α and β are two solutions of the equation a tan x + b sec x = c, then find the values of sin (α + β) and cos (α + β). Watch Solution