TITLE: How Can One Prove $\cos(\pi/7) + \cos(3 \pi/7) + \cos(5 \pi/7) = 1/2$
QUESTION [16 upvotes]: Reference: http://xkcd.com/1047/
We tried various different trigonometric identities. Still no luck. 
Geometric interpretation would be also welcome.
EDIT: Very good answers, I'm clearly impressed. I followed all the answers and they work! I can only accept one answer, the others got my upvote.

REPLY [4 votes]: As a variation of my previous answer, it can be shown (see e.g. this paper) that the eigenvalues of the tridiagonal matrix
$$\begin{pmatrix}1&1&0\\1&0&1\\0&1&0\end{pmatrix}$$
are $2\cos\dfrac{\pi}{7}$, $2\cos\dfrac{3\pi}{7}$, and $2\cos\dfrac{5\pi}{7}$. Since the trace of a matrix is the same as the sum of its eigenvalues,
$$2\cos\dfrac{\pi}{7}+2\cos\dfrac{3\pi}{7}+2\cos\dfrac{5\pi}{7}=1$$
and the identity shown in the OP easily follows.