Answer
$$\displaystyle{\int_{\frac{\pi}{6}}^{\frac{\pi}{3}}{\frac{\sin \theta \cot \theta}{\sec \theta}d\theta}=\frac{\pi}{12}}$$
Work Step by Step
$\displaystyle{I=\int_{\frac{\pi}{6}}^{\frac{\pi}{3}}{\frac{\sin \theta \cot \theta}{\sec \theta}d\theta}\\
I=\int_{\frac{\pi}{6}}^{\frac{\pi}{3}}{\sin \theta \times \frac{\cos \theta}{\sin \theta}\times \cos \theta \ d\theta}\\
I=\int_{\frac{\pi}{6}}^{\frac{\pi}{3}}{\cos ^2\theta \,\,d\theta}\\
I=\frac{1}{2}\int_{\frac{\pi}{6}}^{\frac{\pi}{3}}{\cos 2\theta +1\,\,d\theta}\\
I=\frac{1}{2}\left[ \frac{1}{2}\sin 2\theta +\theta \right] _{\frac{\pi}{6}}^{\frac{\pi}{3}}\\
I=\frac{\pi}{12}}$