$\tan\left(\frac{\pi}{2} -x\right) - \cot\left(\frac{3\pi}{2} -x\right) + \tan(2\pi-x) - \cot(\pi-x) = \frac{4-2\sec^{2}x}{\tan{x}}$
L.S.
$= \cot{x} - \tan{x} - \tan{x} + \cot{x}$
$= 2\cot{x} - 2\tan{x}$
$= 2\left(\frac{\cos{x}}{\sin{x}} - \frac{\sin{x}}{\cos{x}}\right)$
$= 2\left(\frac{\cos^{2}x - \sin^{2}x}{\sin{x}\cos{x}}\right)$
$= 2\left(\frac{1-2\sin^{2}x}{\sin{x}\cos{x}}\right)$
$= \frac{4 - 2\sin^{2}x}{\sin{x}\cos{x}}$
- Not sure where to go from here.