Question: How would you prove the equation$$\small\cos\dfrac {2\pi}{17}=\dfrac {-1+\sqrt{17}+\sqrt{34-2\sqrt{17}}+2\sqrt{17+3\sqrt{17}-\sqrt{34-2\sqrt{17}}-2\sqrt{34+2\sqrt{17}}}}{16}\tag1$$
I'm not too sure how to prove it and I'm not sure where to begin. I started with $\exp(2\pi i)=\cos(2\pi)+i\sin(2\pi)=1\implies \cos(2k\pi)+i\sin(2k\pi)=\exp(2k\pi)$. But I'm not sure what to do next.
The more natural item is $$ 2 \cos \left( \frac{2 \pi}{17} \right), $$ which is one root of $$ x^8 + x^7 - 7 x^6 - 6 x^5 + 15 x^4 + 10 x^3 - 10 x^2 - 4 x + 1, $$ from page 18 of Reuschle (1875) using a method due to Gauss, having to do with cyclotomy, and predating Galois. If $\omega \neq 1$ but $\omega^{17} = 1,$ it is not hard to show that $\omega + \frac{1}{\omega}$ is a root of the given polynomial, using $$ \small 1 + \omega + \omega^2 + \omega^3 + \omega^4 + \omega^5 + \omega^6 + \omega^7 + \omega^8 + \omega^9 + \omega^{10} + \omega^{11} + \omega^{12} + \omega^{13} + \omega^{14} + \omega^{15} + \omega^{16} =0 $$
