# Prove that: (sin 3x + sin x) sin x + (cos 3x – cos x) cos x = 0

Question:

Prove that: $(\sin 3 x+\sin x) \sin x+(\cos 3 x-\cos x) \cos x=0$

Solution:

L.H.S.

$=(\sin 3 x+\sin x) \sin x+(\cos 3 x-\cos x) \cos x$

$=\sin 3 x \sin x+\sin ^{2} x+\cos 3 x \cos x-\cos ^{2} x$

$=\cos 3 x \cos x+\sin 3 x \sin x-\left(\cos ^{2} x-\sin ^{2} x\right)$

$=\cos (3 x-x)-\cos 2 x \quad[\cos (A-B)=\cos A \cos B+\sin A \sin B]$

$=\cos 2 x-\cos 2 x$

$=0$

$=\mathrm{RH} . \mathrm{S} .$