Show that

$$\frac{\cos 2x}{\sin x}+\frac{\sin 2x}{\cos x}\equiv \csc x,x\ne \frac{n\pi }{2},n\in \mathbb{Z}$$

In a study of fluid dynamics, the pressure coefficient $P$ is modeled by the equation

$${\left(\frac{\cos 2\theta }{\sin \theta }+\frac{\sin 2\theta }{\cos \theta }\right)}^2=7-\cot \theta$$

Hence solve this equation for $0<\theta <\pi$, giving your answers to 3 significant figures as appropriate.

Using the result from part (a), or otherwise, find the exact value of

$${\int }_{\frac{\pi }{6}}^{\frac{\pi }{4}}\left(\frac{\cos 2x}{\sin x}+\frac{\sin 2x}{\cos x}\right)\cot xdx$$