<s>[INST] How prove $\cos(\frac{2\pi}{17}) + \cos(\frac{18\pi}{17})+\cos(\frac{26\pi}{17})+\cos(\frac{30\pi}{17}) = \frac{\sqrt{17}-1}{4}$ 
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
 [/INST] 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}
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