Question

Prove:

$\sin 3x+\sin 2x-\sin x=4\sin x\cos \frac{x}{2}\cos \frac{3x}{2}$

Answer 1

$\sin 3x+\sin 2x-\sin x$

$=\sin 3x+2\cos \frac{3x}{2}\sin \frac{x}{2}$

$=2\sin \frac{3x}{2}\cos \frac{3x}{2}+2\cos \frac{3x}{2}\sin \frac{x}{2}$

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

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

$=4\sin x\cos \frac{x}{2}\cos \frac{3x}{2}$

Here proved