A while ago I came across the following identity in an online math forum (of which I don't remember the name): $\tan\left(\frac{\pi}{11}\right)+4\sin\left(\frac{3\pi}{11}\right)=\sqrt{11}.$
It is not hard to give a proof by rewriting everything in terms of $\exp(i\pi/11)$ and applying a sequence of manipulations. I am wondering where this identity is coming from. Can somebody think of a geometric interpretation? Of an algebraic one?
Edit: Here's an example of what I mean by an algebraic interpretation: The identity $\sin\left(\frac{\pi}{7}\right)\cdot\sin\left(\frac{2\pi}{7}\right)\cdot\sin\left(\frac{3\pi}{7}\right)=\frac{\sqrt{7}}{8}$ expresses the fact that for the Chebyshev polynomial $T_7(x)=x(64x^6-112x^4+56x^2-7)$ the product of the roots $\displaystyle \sin\left(\frac{k\pi}{7}\right)$, $1\leq k<7$, of the second factor is equal to the normalized constant term $\displaystyle \frac{7}{64}$.