Answer
$$\frac{22}{105} \sqrt{2}-\frac{8}{105}$$
Work Step by Step
Given $$ $$
Since
\begin{align*}
\int_{\pi / 4}^{\pi / 2} \cot ^{5} \phi \csc ^{3} \phi d \phi&=\int_{\pi / 4}^{\pi / 2} \cot ^{4} \phi \csc ^{2} \phi \csc \phi \cot \phi d \phi\\
&=\int_{\pi / 4}^{\pi / 2}\left(\csc ^{2} \phi-1\right)^{2} \csc ^{2} \phi \csc \phi \cot \phi d \phi
\end{align*}
Let $u=\csc \phi \ \ \to \ \ du=-\csc \phi \cot \phi d \phi$ and at $ \phi =\pi / 4\to u=\sqrt{2}$ at $ \phi =\pi / 2\to u=1$ , then
\begin{align*}
\int_{\pi / 4}^{\pi / 2} \cot ^{5} \phi \csc ^{3} \phi d \phi &=\int_{\pi / 4}^{\pi / 2}\left(\csc ^{2} \phi-1\right)^{2} \csc ^{2} \phi \csc \phi \cot \phi d \phi\\
&=\int_{\sqrt{2}}^{1}\left(u^{2}-1\right)^{2} u^{2}(-d u)\\
&=\int_{1}^{\sqrt{2}}\left(u^{6}-2 u^{4}+u^{2}\right) d u\\
&=\left[\frac{1}{7} u^{7}-\frac{2}{5} u^{5}+\frac{1}{3} u^{3}\right]_{1}^{\sqrt{2}}\\
&=\left(\frac{8}{7} \sqrt{2}-\frac{8}{5} \sqrt{2}+\frac{2}{3} \sqrt{2}\right)-\left(\frac{1}{7}-\frac{2}{5}+\frac{1}{3}\right)\\
&=\frac{22}{105} \sqrt{2}-\frac{8}{105}
\end{align*}