(a)[3]
Prove that $4\sin x\sin\left(x + \tfrac{1}{6}\pi\right) \equiv \sqrt{3} - \sqrt{3}\cos 2x + \sin 2x$.
(b)[4]
Find the exact value of $\displaystyle \int_{0}^{\tfrac{5}{6}\pi} 4\sin x\sin\left(x + \tfrac{1}{6}\pi\right)\, dx$.
(c)[3]
Find the least positive value of $y$ that satisfies the equation $4\sin(2y)\sin\left(2y + \frac{\pi}{6}\right) = \sqrt{3}$. Give your answer exactly.