Evaluation of a trigonometric series $$
\mbox{Question: Evaluate}\quad
\tan^{2}\left(\pi \over 16\right) + \tan^{2}\left(2\pi \over 16\right) + \tan^{2}\left(3\pi \over 16\right) + \cdots + \tan^{2}\left(7\pi \over 16\right)
$$
What I did:
Well I know that $\tan^{2}\left(7\pi/16\right)$ is the same as
$\cot^{2}\left(\pi/16\right)$. Thus this will repeat for all values up to
$\tan^{2}\left(4\pi/16\right)$.
However, I don't understand where to proceed from there. 

The 15 numbers $\tan k\pi/16$, $-7\leq k\leq7$, are the roots of the polynomial (of degree 15)
$(1+ix)^{16}-(1-ix)^{16}$ (using the geometric interpretation of complex numbers and their multiplication). The polynomial is of the form $-32ixp(x^2)$, where $p(t)=t^7-35t^6+\dots$ is a polynomial of degree 7, the roots of $p$ are thus $\tan^2 k\pi/2$, $1\leq k\leq7$. Your sum  is therefore $35$.
