Boundary Values & Fourier Series · Superchallenge problem

Show that $\ln\!\left(2\sin\dfrac x2\right) = -\displaystyle\sum_{n\ge1}\dfrac{\cos nx}{n}$ for $0 < x < 2\pi$, and use Parseval's identity to evaluate $\displaystyle\int_0^{\pi}\ln^2\!\left(2\sin\dfrac x2\right)dx$.