Answer
$$\int_{\frac{\pi}{6}}^{\frac{\pi}{2}}cot^{2}x\,dx=\sqrt{3}-\frac{\pi}{3}$$
Work Step by Step
$$cot^{2}x=csc^{2}x-1$$
$$\int_{\frac{\pi}{6}}^{\frac{\pi}{2}}cot^{2}x\,dx=\int_{\frac{\pi}{6}}^{\frac{\pi}{2}}(csc^{2}x-1)dx$$
$$=\left [ -cotx-x \right ]_{\pi/6}^{\pi/2}$$
$$=(0-\frac{\pi}{2})-(-\sqrt{3}-\frac{\pi}{6})$$
$$=\sqrt{3}-\frac{\pi}{3}$$
