Prove that

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

Taking LHS,

$=\frac{1-\cos 2 x+\sin x}{\sin 2 x+\cos x}$

$=\frac{(1-\cos 2 x)+\sin x}{\sin 2 x+\cos x}$

We know that,

$1-\cos 2 x=2 \sin ^{2} x \& \sin 2 x=2 \sin x \cos x$

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

Taking sinx common from the numerator and cosx from the denominator

$=\frac{\sin x(2 \sin x+1)}{\cos x(2 \sin x+1)}$

$=\frac{\sin x}{\cos x}$

$=\tan \mathrm{x}\left[\because \tan \theta=\frac{\sin \theta}{\cos \theta}\right]$

= RHS

∴ LHS = RHS

Hence Proved