Find $S$ if $S=\tan^21^\circ+\tan^23^\circ+\tan^25^\circ+\dots+\tan^289^\circ$.
I tried converting all tangents above $45^\circ$ to cotangents and added them with tangents with the same angle, but that does not give any result nor does it telescope. Tried writing $\tan^2\theta = 1-\frac{\tan\theta}{\tan2\theta}$, but again, no headway. I am trying to relate this problem with complex numbers, and saying roots of a polynomial are of the form $\tan[(2n+1)\frac{\pi}{180}]$, but haven't been able to do so.
