Answer
$\frac{1+\cos 2x}{\sin 2x}=\cot x$
Work Step by Step
Start with the left side:
$\frac{1+\cos 2x}{\sin 2x}$
Use the double-angle identities $\cos 2x=2\cos^2 x-1$ and $\sin 2x=2\sin x\cos x$:
$=\frac{1+2\cos^2 x-1}{2\sin x\cos x}$
Simplify:
$=\frac{2\cos^2 x}{2\sin x\cos x}$
$=\frac{\cos x}{\sin x}$
$=\cot x$
Since this equals the right side, the identity has been proven.