Problem source

Show, using de Moivre's theorem, or otherwise, that 
\[
\tan7\theta=\frac{t(t^{6}-21t^{4}+35t^{2}-7)}{7t^{6}-35t^{4}+21t^{2}-1}\,,
\]
where $t=\tan\theta.$
\begin{questionparts}
\item By considering the equation $\tan7\theta=0,$ or otherwise, obtain
a cubic equation with integer coefficients whose roots are 
\[
\tan^{2}\left(\frac{\pi}{7}\right),\ \tan^{2}\left(\frac{2\pi}{7}\right)\ \mbox{ and }\tan^{2}\left(\frac{3\pi}{7}\right)
\]
and deduce the value of 
\[
\tan\left(\frac{\pi}{7}\right)\tan\left(\frac{2\pi}{7}\right)\tan\left(\frac{3\pi}{7}\right)\,.
\]
\item Find, without using a calculator, the value of 
\[
\tan^{2}\left(\frac{\pi}{14}\right)+\tan^{2}\left(\frac{3\pi}{14}\right)+\tan^{2}\left(\frac{5\pi}{14}\right)\,.
\]
\end{questionparts}