Prove that

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

Proof: consider, L.H.S $=\frac{\sin x}{1+\cos x}$

$\frac{\sin x}{1+\cos x}=\frac{2 \cos \frac{x}{2} \sin \frac{x}{2}}{1+\cos ^{2} \frac{x}{2}-\sin ^{2} \frac{x}{2}}\left(\because \cos ^{2} \frac{x}{2}-\sin ^{2} \frac{x}{2}=\cos x\right.$ and $\left.2 \cos \frac{x}{2} \sin \frac{x}{2}=\sin x\right)$

$=\frac{2 \cos \frac{x}{2} \sin \frac{x}{2}}{\cos ^{2} \frac{x}{2}+\sin ^{2} \frac{x}{2}+\cos ^{2} \frac{x}{2}-\sin ^{2} \frac{x}{2}}\left(\because \cos ^{2} \frac{x}{2}-\sin ^{2} \frac{x}{2}=1\right)$

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

$\frac{\sin x}{1+\cos x}=\tan \frac{x}{2}=$ R.H.S

Since L.H.S = R.H.S, Hence proved.