Question

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

Answer 1

$\frac{1+\cos x}{\sin x}=\cot \frac{x}{2}$

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

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

$L.H.S=\frac{1+\cos x}{\sin x}$

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

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

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

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

$\therefore$ $L.H.S=R.H.S$

$\therefore$ $\frac{1+\cos x}{\sin x}=\cot \frac{x}{2}$