Statement - 1: $\int\limits_0^{\pi / 2} x \cot x d x=\frac{\pi}{2} \log 2$

Statement - 2: $\int\limits_0^{\pi / 2} \log \sin x d x=-\frac{\pi}{2} \log 2$

Statement-1 is True, Statement-2 is True; Statement-2 is a correct explanation for Statement-1.

Clearly, statement-2 is true.

Now, $\int\limits_0^{\pi / 2} x \cot x d x=[x \log \sin x]_0^{\pi / 2}-\int\limits_0^{\pi / 2} \log \sin x d x$

$\Rightarrow \int\limits_0^{\pi / 2} x \cot x d x=0-\lim _{x \rightarrow 0} x \log \sin x-\left(-\frac{\pi}{2} \log 2\right)$

$\Rightarrow \int\limits_0^{\pi / 2} x \cot x d x=\frac{\pi}{2} \log 2$

So, statement-1 is true. Also, Statement-2 is a correct explanation for statement.