Answer
$$0$$
Work Step by Step
$$\eqalign{
& \int_{\pi /4}^{3\pi /4} {\csc z\cot z} dz \cr
& {\text{We know that}}\cr
&\,\,\,\frac{d}{{d\theta }}\left[ {\csc \theta } \right] = - \csc \theta \cot \theta \cr
&{\text{Then}}{\text{,}} \cr
& \int_{\pi /4}^{3\pi /4} {\csc z\cot z} dz = - \left( {\csc z} \right)_{\pi /4}^{3\pi /4} \cr
& {\text{Evaluate}} \cr
& = - \left( {\csc \frac{{3\pi }}{4} - \csc \frac{\pi }{4}} \right) \cr
& {\text{Simplifying, we get:}} \cr
& = - \left( {\sqrt 2 - \sqrt 2 } \right) \cr
& = 0 \cr} $$
