# How do you prove that  cot^2x = (1+cos2x)/(1-cos2x) ?

$\cos 2 x = \frac{1 - {\tan}^{2} x}{1 + {\tan}^{2} x}$
$\implies \frac{1 + \cos 2 x}{1 - \cos 2 x} = \frac{1 + \left(\frac{1 - {\tan}^{2} x}{1 + {\tan}^{2} x}\right)}{1 - \left(\frac{1 - {\tan}^{2} x}{1 + {\tan}^{2} x}\right)}$
$\frac{1 + {\tan}^{2} x + 1 - {\tan}^{2} x}{1 + {\tan}^{2} x - 1 + {\tan}^{2} x}$
$= \frac{2}{2 {\tan}^{2} x} = {\cot}^{2} x$