Question

Prove the following identity:
$\frac{1-\cos 2x+\sin 2x}{1+\cos 2x+\sin 2x}=\tan x$

Answer 1

To prove: $\frac{1-\cos 2x+\sin 2x}{1+\cos 2x+\sin 2x}=\tan x$

L.H.S.
$=\frac{1-\cos 2x+\sin 2x}{1+\cos 2x+\sin 2x}$

$=\frac{2{\sin }^{2}x+\sin 2x}{2{\cos }^{2}x+\sin 2x}\left[\begin{array}{c}\because \cos 2x=1-2{\sin }^{2}x\\ \text{and}\cos 2x=2{\cos }^{2}x-1\end{array}\right]$

$=\frac{2{\sin }^{2}x+2\sin x\cos x}{2{\cos }^{2}x+2\sin x\cos x}(\because \sin 2x=2\sin x\cos x$

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

$=\tan x$
=R.H.S.
Hence proved.