Question

Prove the identity: (sin 3x + sin x)sin x + (cos 3x – cos x)cos x = 0

Answer 1

Let us consider the LHS

(sin 3x + sin x) sin x + (cos 3x – cos x) cos x

= (sin 3x) (sin x) + sin² x + (cos 3x) (cos x) – cos² x

= [(sin 3x) (sin x) + (cos 3x) (cos x)] + (sin² x – cos² x)

= [(sin 3x) (sin x) + (cos 3x) (cos x)] – (cos² x – sin² x)

= cos (3x – x) – cos 2x

As we know, cos 2x = cos² x – sin² x

cos A cos B + sin A sin B = cos(A – B)

Therefore,

= cos 2x – cos 2x

= 0

= RHS

Thus proved.