Answer
$\displaystyle \frac{12}{35}$
Work Step by Step
Let $\left[\begin{array}{ll}
\theta=\cot^{-1}u & u=\cot\theta\\
& du=-\csc^{2}\theta d\theta
\end{array}\right], (\cot^{2}t=\csc^{2}t-1)$
$\displaystyle \int\csc^{4}\theta\cot^{4}\theta d\theta=\int\csc^{2}\theta\cot^{4}\theta(\csc^{2}\theta d\theta)=$
$=\displaystyle \int(u^{2}+1)u^{4}(-du)$
$=-\displaystyle \int u^{6}du-\int u^{4}du$
$=-\displaystyle \frac{u^{7}}{7}-\frac{u^{5}}{5}+C$
... bring back theta
$=-\displaystyle \frac{\cot^{7}\theta}{7}-\frac{\cot^{5}\theta}{5}+C$
$\displaystyle \int_{\pi/4}^{\pi/2}\csc^{4}\theta\cot^{4}\theta d\theta=\left|-\frac{\cot^{7}\theta}{7}-\frac{\cot^{5}\theta}{5}\right|_{\pi/4}^{\pi/2}$
$=0-(-\displaystyle \frac{1}{7}-\frac{1}{5})$
$=\displaystyle \frac{12}{35}$
