Answer
See the steps.
Work Step by Step
$LHS =\tan{\dfrac{x}{2}}-\cot{\dfrac{x}{2}}= \dfrac{1-\cos{x}}{\sin{x}} - \dfrac{\sin{x}}{1-\cos{x}}$
$LHS =\dfrac{1-2\cos{x}+\cos^2{x}-\sin^2{x}}{\sin{x}(1-\cos{x})} = \dfrac{-2\cos{x}+2\cos^2{x}}{\sin{x}(1-\cos{x})}$
$LHS = \dfrac{-2\cos{x}(1-\cos{x})}{\sin{x}(1-\cos{x})} = \dfrac{-2\cos{x}}{\sin{x}} = -2 \cot{x} = RHS$
