(i)[4]
Show that $\cosec 2x + \cot 2x \equiv \cot x$ by simplifying the left-hand side.
(ii)[2]
Hence find the exact value of $\cot \frac{\pi}{12}$ from the result above.
(iii)[3]
Find $\displaystyle \int \sin 2x(\cosec 4x + \cot 4x)\,dx$ by simplifying the integrand first.