Question

Prove that $\frac{\sin x}{1+\cos x}=\tan \frac{x}{2}.$

Answer 1

L.H.S

$\frac{\sin x}{1+\cos x}$

We know that

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

$\cos x=2{\cos }^2\frac{x}{2}-1$

Therefore,

$=\frac{2\sin \frac{x}{2}.\cos \frac{x}{2}}{1+2{\cos }^2\frac{x}{2}-1}$

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

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

$=\tan \frac{x}{2}$

R.H.S

Hence, proved.