Show that $\int_0^\pi \log^2\left(\tan\frac{ x}{4}\right)dx=\frac{\pi^3}{4}.$
The following Sum Identity was derived by applying Parseval's Theorem to the integral in the question linked above.
$\displaystyle \large \sum_{k=1}^\infty \left(\frac{2\mathcal{H}_{2k} - \mathcal{H}_k}{k} \right)^2 = \frac{\pi^4}{8}$
Is there a means of evaluating the sum without invoking Parseval's Theorem?
