Prove that

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

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

$=(2 \sin 2 x \cos x) \sin x-(2 \sin 2 x \sin x) \cos x$

$=0 .$

Using the formula,

$\sin A+\sin B=2 \sin \frac{A+B}{2} \cos \frac{A-B}{2}$

$\cos A-\cos B=-2 \sin \frac{A+B}{2} \sin \frac{A-B}{2}$