How to prove an identity (Trigonometry Angles--Pi/13) In this page http://mathworld.wolfram.com/TrigonometryAnglesPi13.html
I found equation (11) and (12).
$$\cos^2\frac{\pi}{13}+\cos^2\frac{3\pi}{13}+\cos^2\frac{4\pi}{13}=\frac{11+\sqrt{13}}{8}$$
$$\sin\frac{\pi}{13}+\sin\frac{3\pi}{13}+\sin\frac{4\pi}{13}=\sqrt{\frac{13+3\sqrt{13}}{8}}$$
How to prove it ?
Thanks in advances

Since $\pm 1,\pm 3,\pm 4$ are the only quadratic residues in $\mathbb{F}_{13}^*$, both sums can be related with a quadratic Gauss sum $\pmod{13}$. You just need to apply the cosine duplication formula to the first one and to consider the square of the second one.
