$a)\quad \dfrac{\tan x}{\sin x} -\dfrac{\sin x}{\cot x}$
$=\dfrac{\dfrac{\sin x}{\cos x}}{\sin x} -\dfrac{\sin x}{\dfrac{\cos x}{\sin x}}$
$=\dfrac{1}{\cos x} -\dfrac{\sin^2x}{\cos x}$
$=\dfrac{1 -\sin^2x}{\cos x}$
$=\dfrac{\cos^2x}{\cos x}$
$=\cos x$
$b)\quad \dfrac{1 -2\sin^2x}{1 + 2\sin x\cos x}$
$=\dfrac{\cos^2x +\sin^2x -2\sin^2x}{\sin^2x +2\sin x\cos x +\cos^2x}$
$=\dfrac{\cos^2x -\sin^2x}{(\sin x +\cos x)^2}$
$=\dfrac{(\cos x -\sin x)(\cos x +\sin x)}{(\sin x +\cos x)^2}$
$=\dfrac{\cos x -\sin x}{\cos x +\sin x}$
$=\dfrac{\dfrac{\cos x -\sin x}{\cos x}}{\dfrac{\cos x +\sin x}{\cos x}}$
$=\dfrac{1 -\tan x}{1 +\tan x}$
$c)\quad \cot^2x -\cos^2x$
$=\dfrac{\cos^2x}{\sin^2x} - \cos^2x$
$=\cos^2x\left(\dfrac{1}{\sin^2x} -1\right)$
$=\cos^2x\cdot\dfrac{1 -\sin^2x}{\sin^2x}$
$=\cos^2x\cdot\dfrac{\cos^2x}{\sin^2x}$
$=\cos^2x.\cot^2x$
