Answer
$$2$$
Work Step by Step
$$\eqalign{
& \int_{\pi /4}^{3\pi /4} {{{\csc }^2}x} dx \cr
& {\text{Recall that }}\cr
&\frac{d}{{dx}}\left[ {\cot x} \right] = - {\csc ^2}x\cr
&{\text{ Then}}{\text{,}} \cr
& \int_{\pi /4}^{3\pi /4} {{{\csc }^2}x} dx = - \left( {\cot x} \right)_{\pi /4}^{3\pi /4} \cr
& {\text{Evaluate}} \cr
& = - \left( {\cot \left( {\frac{{3\pi }}{4}} \right) - \cot \left( {\frac{\pi }{4}} \right)} \right) \cr
& {\text{Simplify}} \cr
& = - \left( { - 1 - 1} \right) \cr
& = 2 \cr} $$
