Answer
$\frac{2\cot x}{\tan 2x}=\csc^2 x-2$
Work Step by Step
Start with the left side:
$\frac{2\cot x}{\tan 2x}$
Simplify using the Double-Angle Formula for tangent:
$=\frac{2\cot x}{\frac{2\tan x}{1-\tan^2 x}}$
Multiply the top and bottom by $1-\tan^2 x$:
$=\frac{2\cot x}{\frac{2\tan x}{1-\tan^2 x}}*\frac{1-\tan^2 x}{1-\tan^2 x}$
Simplify:
$=\frac{2\cot x(1-\tan^2 x)}{2\tan x}$
$=\frac{\cot x(1-\tan^2 x)}{\frac{1}{\cot x}}$
Multiply the top and bottom by $\cot x$:
$=\frac{\cot x(1-\tan^2 x)}{\frac{1}{\cot x}}*\frac{\cot x}{\cot x}$
Simplify:
$=\cot^2 x(1-\tan^2 x)$
$=\cot^2 x-1$
$=(\csc^2 x-1)-1$
$=\csc^2 x-2$
Since this equals the right side, the identity has been proven.
