Prove the following: sin (n + 1)x sin (n + 2)x + cos (n + 1)x cos (n + 2)x = cos x

Solution:

LHS = 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

= cos [(n + 2)x - (n + 1)x] 

 [By trigonometric formulas, cos (A - B) = cos A cos B + sin A sin B]

= cos [(n + 2 - n - 1)x]

= cos x

= RHS

NCERT Solutions Class 11 Maths Chapter 3 Exercise 3.3 Question 10

Prove the following: sin (n + 1)x sin (n + 2)x + cos (n + 1)x cos (n + 2)x = cos x

Summary:

We got, sin (n + 1)x sin (n + 2)x + cos (n + 1)x cos (n + 2)x = cos x. Hence Proved