#### Answer

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

#### Work Step by Step

Start with the left side:
$2\cos^2 \frac{x}{2}\tan x$
Use the half-angle identity for cosine:
$=2\left(\pm\sqrt{\frac{1+\cos x}{2}}\right)^2\tan x$
Simplify:
$=2*\frac{1+\cos x}{2}*\tan x$
$=(1+\cos x)\tan x$
$=\tan x+\cos x\tan x$
$=\tan x+\cos x*\frac{\sin x}{\cos x}$
$=\tan x+\sin x$
Since this equals the right side, the identity has been proven.