Question

Prove the following:

sin (n+1)x sin (n+2)x + cos (n+1)x cos (n+2)x = cos x

Answer 1

We have

L.H.S = sin (n+1)x sin (n+2)x + cos (n+1)x cos (n+2)x = cos x

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

[$\because$ cos(A-b) =cos A cos B + sin A sin B]

= cos [ nx+x - nx -2x]

= cos (-x) = cos x = R.H.S

[$\because$ cos $(-\theta )$ = cos $\theta$]