Answer
$\frac{2~cot~x}{tan~2x} = csc^2~x-2$
Work Step by Step
$\frac{2~cot~x}{tan~2x} = \frac{2~cos~x~cos~2x}{sin~x~sin~2x}$
$\frac{2~cot~x}{tan~2x} = \frac{2~cos~x~(cos^2~x-sin^2~x)}{sin~x~2~sin~x~cos~x}$
$\frac{2~cot~x}{tan~2x} = \frac{cos^2~x-sin^2~x}{sin^2~x}$
$\frac{2~cot~x}{tan~2x} = \frac{(1-sin^2~x)-sin^2~x}{sin^2~x}$
$\frac{2~cot~x}{tan~2x} = \frac{1-2~sin^2~x}{sin^2~x}$
$\frac{2~cot~x}{tan~2x} = \frac{1}{sin^2~x}-\frac{2~sin^2~x}{sin^2~x}$
$\frac{2~cot~x}{tan~2x} = csc^2~x-2$
