Answer
$$ - \frac{{{{\left( {\cot x} \right)}^2}}}{2} + C$$
Work Step by Step
$$\eqalign{
& \int {{{\csc }^2}x\cot xdx} \cr
& {\text{substitute }}u = \cot x,{\text{ }}du = - {\csc ^2}xdx \cr
& = \int {{{\csc }^2}x\cot xdx} = \int {u\left( { - du} \right)} \cr
& = - \int u du \cr
& {\text{find the antiderivative}} \cr
& = - \frac{{{u^2}}}{2} + C \cr
& {\text{replacing }}u = \cot x \cr
& = - \frac{{{{\left( {\cot x} \right)}^2}}}{2} + C \cr} $$
