Question

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

Answer 1

$$\begin{gathered} \mathrm{LHS}=\sqrt{\frac{1-\cos 2x}{1+\cos 2x}} \\ =\sqrt{\frac{2{\sin }^{2}x}{2{\cos }^{2}x}}\left[\because 1-\cos 2x=2{\sin }^{2}x\mathrm{and}1+\cos 2x=2{\cos }^{2}x\right] \\ =\frac{\sin x}{\cos x} \\ =\tan x=\mathrm{RHS} \\ \mathrm{Hence}\mathrm{proved}. \end{gathered}$$