Prove that

To Prove: $\cot \frac{x}{2}-\tan \frac{x}{2}=2 \cot x$

Proof: Consider L.H.S,

$\cot \frac{x}{2}-\tan \frac{x}{2}=\frac{\cos \frac{x}{2}}{\sin \frac{x}{2}}-\frac{\sin \frac{x}{2}}{\cos \frac{x}{2}}$

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

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

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

Here multiply and divide L.H.S by 2

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

$=\frac{2 \cos x}{\sin x}(\because 2 \sin x \cos x=\sin 2 x)$

$\Rightarrow\left(2 \sin \frac{x}{2} \cos \frac{x}{2}=\sin x\right)$

$\cot -\tan \frac{x}{2}=2 \cot x=$ R.H.S

$\therefore$ L.H.S $=$ R.H.S, Hence proved