Answer
$$\frac{2}{3}(3 \sqrt{3}-1)$$
Work Step by Step
Given
$$ \int_{\pi / 3}^{\pi / 2} \cot ^{2} \frac{x}{2} \csc ^{2} \frac{x}{2} d x$$
Let
$$ u= \cot \frac{x}{2} \ \ \ \Rightarrow \ \ \ du =-\frac{1}{2} \csc ^{2} \frac{x}{2} d x$$
At $$ x= \pi/3 \to u= \sqrt{3}, \ \ x= \pi/2\to u= 1$$
Then
\begin{aligned}\int_{\pi / 3}^{\pi / 2} \cot ^{2} \frac{x}{2} \csc ^{2} \frac{x}{2} d x&=-2 \int_{\sqrt{3}}^{1} u^{2} d u\\
&=-\left.\frac{2}{3} u^{3}\right|_{\sqrt{3}} ^{1}\\
&=\frac{2}{3}(3 \sqrt{3}-1)\\ \end{aligned}
