It is known that the quantity $\cos \frac{2π}{17}$ is a root of the $8$'th degree equation, $$x^8 + \frac{1}{2} x^7 - \frac{7}{4} x^6 - \frac{3}{4} x^5 + \frac{15}{16} x^4 + \frac{5}{16} x^3 - \frac{5}{32} x^2 - \frac{x}{32} + \frac{1}{256} = 0$$ It is known that the regular $17$ sided polygon can be constructed from $cos \frac{2π}{17}$ , if this can be expressed in square roots. Can you show that this equation can be derived from the relation $9\theta=-\sin 8\theta$, where $\theta = \frac{2π}{17}$?
Can you demonstrate that it is possible to solve this particular $8$'th degree equation, even though there is no $8$'th degree formula? ( there is more than one possible form of solution; it is known that there is often more than one expression in radicals for the same quantity) Also, can you find a square root form for $\cos \frac{2\pi}{17}$ which has minimum number of terms?
$$\cos\frac{2\pi}{17}=\tfrac{-1+\sqrt{17}+\sqrt{34-2\sqrt{17}}+2\sqrt{17+3\sqrt{17}-\sqrt{34-2\sqrt{17}}-2\sqrt{34+2\sqrt{17}}}}{16}$$ See here: https://www.wolframalpha.com/input/?i=-cos%282pi%2F17%29%2B%28-1%2B%5Csqrt%7B17%7D%2B%5Csqrt%7B34-2%5Csqrt%7B17%7D%7D%2B2%5Csqrt%7B17%2B3%5Csqrt%7B17%7D-%5Csqrt%7B34-2%5Csqrt%7B17%7D%7D-2%5Csqrt%7B34%2B2%5Csqrt%7B17%7D%7D%7D%29%2F16