Question

Prove that
$\frac{sin7\theta -sin5\theta }{cos7\theta +cos5\theta }=tan6\theta$

Answer 1

L.H.S$=\frac{\sin 7\theta -\sin 5\theta }{\cos 7\theta +\cos 5\theta }$

using transformation angle formula,

$=\frac{2\cos \left(\frac{7\theta -5\theta }{2}\right)\sin \left(\frac{7\theta +5\theta }{2}\right)}{2\cos \left(\frac{7\theta -5\theta }{2}\right)\cos \left(\frac{7\theta +5\theta }{2}\right)}$

$=\frac{2\cos \left(\frac{2\theta }{2}\right)\sin \left(\frac{12\theta }{2}\right)}{2\cos \left(\frac{2\theta }{2}\right)\cos \left(\frac{12\theta }{2}\right)}$

$=\frac{\cos \theta \sin 6\theta }{\cos \theta \cos 6\theta }$

$=\frac{\sin 6\theta }{\cos 6\theta }$

$=\tan 6\theta =$R.H.S

Hence proved.