Answer
$${\sin ^2}x\left( {1 + \cot x} \right) + {\cos ^2}x\left( {1 - \tan x} \right) + {\cot ^2}x = {\csc ^2}x$$
Work Step by Step
$$\eqalign{
& {\sin ^2}x\left( {1 + \cot x} \right) + {\cos ^2}x\left( {1 - \tan x} \right) + {\cot ^2}x = {\csc ^2}x \cr
& {\text{We transform the more complicated left side to match the right side}}. \cr
& = {\sin ^2}x + \sin x\cos x + {\cos ^2}x - \sin x\cos x + {\cot ^2}x \cr
& = \left( {{{\sin }^2}x + {{\cos }^2}x} \right) + {\cot ^2}x \cr
& = 1 + {\cot ^2}x \cr
& = {\csc ^2}x \cr
& {\text{Thus have verified that the given equation is an identity}} \cr} $$
