Question

Prove that $\sin x+\sin 3x+\sin 5x+\sin 7x=4\cos x\cos 2x\sin 4x$

Answer 1

$L.H.S=\sin x+\sin 3x+\sin 5x+\sin 7x$
$=(\sin 7x+\sin x)+(\sin 5x+\sin 3x)$
$=\left\{2\sin \left(\frac{7x+x}{2}\right)\cos \left(\frac{7x-x}{2}\right)\right\}+\left\{2\sin \left(\frac{5x+3x}{2}\right)\cos \left(\frac{5x-3x}{2}\right)\right\}$
$\because \sin A+\sin B=2\sin \left(\frac{A+B}{2}\right)\cos \left(\frac{A-B}{2}\right)=2\sin 4x\cos 3x+2\sin 4x\cos x$
$=2\sin 4x[\cos 3x+\cos x]$
$=2\sin 4x\left\{2\cos \left(\frac{3x+x}{2}\right)\cos \left(\frac{3x-x}{2}\right)\right\}$
$=2\sin 4x\left(2\cos 2x\cos x\right)$
$=4\cos x\cos 2x\sin 4x=R.H.S.$

Hence proved.