Prove that \[ \sqrt{\frac{1-\cos 2x}{1+\cos 2x}}=\tan x \]
Proof:
\[
LHS=\sqrt{\frac{1-\cos 2x}{1+\cos 2x}}
\]
Using the identities:
\[
1-\cos 2x=2\sin^2 x
\]
\[
1+\cos 2x=2\cos^2 x
\]
Substituting these values:
\[
LHS=\sqrt{\frac{2\sin^2 x}{2\cos^2 x}}
\]
\[
=\sqrt{\frac{\sin^2 x}{\cos^2 x}}
\]
\[
=\sqrt{\tan^2 x}
\]
\[
=\tan x
\]
Hence proved,
\[
\boxed{\sqrt{\frac{1-\cos 2x}{1+\cos 2x}}=\tan x}
\]