(a)[4]
Prove that $\sin 2\theta (a \cot \theta + b \tan \theta) \equiv a + b + (a - b) \cos 2\theta$, where $a$ and $b$ are constants.
(b)[3]
Find the exact value of $\displaystyle \int_{\frac{1}{12}\pi}^{\frac{1}{6}\pi} \sin 2\theta (5\cot \theta + 3 \tan \theta)\, d\theta$.
(c)[3]
Solve the equation $\sin^2\left(\frac{2}{3}\alpha\right)(2 \cot \tfrac{1}{3}\alpha + 7 \tan \tfrac{1}{3}\alpha) = 11$ for $-\pi < \alpha < \pi$.