Answer
$$1$$
Work Step by Step
$$\eqalign{
& \int_{\pi /4}^{\pi /2} {{{\csc }^2}\theta } d\theta \cr
& {\text{integrate using the basic rules for integration}} \cr
& = \left. {\left( { - \cot \theta } \right)} \right|_{\pi /4}^{\pi /2} \cr
& {\text{using the fundamental theorem}} \cr
& = - \left( {\cot \frac{\pi }{2} - \cot \frac{\pi }{4}} \right) \cr
& {\text{simplify}} \cr
& = - \left( {0 - 1} \right) \cr
& = 1 \cr} $$
