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