Answer
$$\frac{12}{35}$$
Work Step by Step
Given $$\int_{\pi / 4}^{\pi / 2} \csc ^{4} \theta \cot ^{4} \theta d \theta$$
Since
\begin{align*}
\int_{\pi / 4}^{\pi / 2} \csc ^{4} \theta \cot ^{4} \theta d \theta
&=\int_{\pi / 4}^{\pi / 2} \csc ^{2} \theta\csc ^{2} \theta \cot ^{4} \theta d \theta\\
&=\int_{\pi / 4}^{\pi / 2} \csc ^{2} \theta(1+ \cot ^{2} \theta) \cot ^{4} \theta d \theta\\
&=\int_{\pi / 4}^{\pi / 2} \csc ^{2} \theta(\cot ^{4} \theta+ \cot ^{6} \theta) d \theta\\
&=\frac{1}{5}\cot ^{5} \theta+\frac{1}{7} \cot ^{7} \theta\bigg|_{\pi/4}^{\pi/2}\\
&=\frac{12}{35}
\end{align*}