How prove $\cos(\frac{2\pi}{17}) + \cos(\frac{18\pi}{17})+\cos(\frac{26\pi}{17})+\cos(\frac{30\pi}{17}) = \frac{\sqrt{17}-1}{4}$

Question

Prove that $\cos(\frac{2\pi}{17}) + \cos(\frac{18\pi}{17})+\cos(\frac{26\pi}{17})+\cos(\frac{30\pi}{17}) = \frac{\sqrt{17}-1}{4}$

Regards that value of $\cos(2\pi/17)$, I can't find the easy way to solve that expression.

Even if I had time, I wouldn't try that method to find the all roots others cosines expressions. IMHO

Answer 1

Let $p$ be an odd prime number. Then $$g_p=\sum_{k=0}^{p-1}\exp(2\pi i k^2/p)$$ is a quadratic Gauss sum. Gauss proved that $g_p=\sqrt p$ or $i\sqrt p$ according to whether $p\equiv1$ or $p\equiv3\pmod 4$. It is quite easy to prove this up to sign, but hard to prove the sign.

So $g_{17}=\sqrt{17}$. Therefore \begin{align} \sqrt{17}&=1+2\exp(2\pi i/17)+2\exp(8\pi i/17)+2\exp(18\pi i/17) +2\exp(32\pi i/17)\\ &+2\exp(16\pi i/17)+2\exp(4\pi i/17)+2\exp(30\pi i/17) +2\exp(26\pi i/17)\\ &=1+4\cos(2\pi/17)+4\cos(18\pi/17)+4\cos(26\pi/17)+4\cos(30\pi/17). \end{align}