(a)[4]
Prove the identity $\sin 2x(\cot x + 3\tan x) = 4 - 2\cos 2x$.
(b)[2]
Hence determine the exact value of $\cot \frac{\pi}{12} + 3\tan \frac{\pi}{12}$.
(c)[5]
The diagram displays the curve with equation $y = 4 - 2\cos 2x$, for $0 \le x \le 2\pi$. At point $A$, the gradient of the curve is $4$. Point $B$ is a minimum point. The $x$-coordinates of $A$ and $B$ are $a$ and $b$ respectively. Show that $\int_a^b (4 - 2\cos 2x)\,dx = 3\pi + 1$.