Question

Prove that $\sqrt{\frac{1-\cos x}{1+\cos x}}=\tan \frac{x}{2}$

Answer 1

As $\cos x=1-2{\sin }^{2}x/2=2{\cos }^{2}x/2-1$

$\Rightarrow 1-\cos x=2{\sin }^{2}x/2,1+\cos x=2{\cos }^{2}x/2$

$\Rightarrow \sqrt{\frac{1-\cos x}{1+\cos x}}=\sqrt{\frac{2{\sin }^{2}x/2}{2{\cos }^{2}x/2}}=\sqrt{{\tan }^{2}x/2}=\tan x/2$.