$\begin{align}  & \int{\frac{\sin 2\theta }{\sec 2\theta }d\theta }=\int{\left( \sin 2\theta \times \frac{1}{\tfrac{1}{\cos 2\theta }} \right)d\theta } \\ & \int{\frac{\sin 2\theta }{\sec 2\theta }d\theta }=\int{\sin 2\theta \cos 2\theta d\theta } \\ & \int{\frac{\sin 2\theta }{\sec 2\theta }d\theta }=\int{2\sin 4\theta d\theta }=\frac{1}{2}\left[ \frac{-\cos 4\theta }{4} \right]+C \\ & \int{\frac{\sin 2\theta }{\sec 2\theta }d\theta }=-\frac{\cos 4\theta }{8}+C \\\end{align}$