Prove that: $(\sin 3 x+\sin x) \sin x+(\cos 3 x-\cos x) \cos x=0$
Prove that: $(\sin 3 x+\sin x) \sin x+(\cos 3 x-\cos x) \cos x=0$
$L.H.S.$ $=(\sin 3 x+\sin x) \sin x+(\cos 3 x-\cos x) \cos x$
$=\sin 3 x \sin x+\sin ^{2} x+\cos 3 x \cos x-\cos ^{2} x$
$=\cos 3 x \cos x+\sin 3 x \sin x-\left(\cos ^{2}-\sin ^{2} x\right)$
$=\cos (3 x-x)-\cos 2 x \quad[\cos (A-B)=\cos A \cos B+\sin A \sin B]$
$=\cos 2 x-\cos 2 x$
$=0$
$= R . H.S.$
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