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