Answer
$$\int_{\frac{\pi}{6}}^{\frac{\pi}{2}}cot^{2}x\,dx=\sqrt{3}-\frac{\pi}{3}$$
Work Step by Step
$1+cot^{2}x=csc^{2}x,\,\,\int csc^{2}x\,dx=-cotx+C$
So$$\int_{\frac{\pi}{6}}^{\frac{\pi}{2}}cot^{2}x\,dx=\int_{\frac{\pi}{6}}^{\frac{\pi}{2}}(csc^{2}x-1)dx$$
$$=\left |-cotx-x \right |_{\frac{\pi}{6}}^{\frac{\pi}{2}}$$
$$=0-\frac{\pi}{2}+\sqrt{3}+\frac{\pi}{6}$$
$$=\sqrt{3}-\frac{\pi}{3}$$