<s>[INST] Radical expression for Cosine formulas Is there nice radical expression for $$\cos\left(\frac{\pi}{2^k+1}\right)?$$
Example: $\cos\left(\dfrac{\pi}{5}\right)=\dfrac{\sqrt{5}+1}{4}$.
Please provide some concrete examples.
Also please provide a general procedure.
I would like to handle $$\cos\left(\frac{a\pi}{2^k+1}\right)$$ as well for $a\in\{0,1,2,\dots,k-1\}$.
 [/INST] For the form $p=2^k+1$, you can find an expression in real (not complex) radicals for,
$$\cos\Big(\frac{n\pi}{p}\Big)$$
and $n$ a positive integer only when $p$ is a Fermat prime, hence $p=3,5,17,257,65537$.  For higher $p$ there can be several ways to express it. 
I. For $p=17$:
$$\cos\frac{2\pi}{17}=\frac{1}{4}\Bigl(\frac{1}{x}+\sqrt{x}\,(17+4\sqrt{17})^{1/4}\Bigr),\quad x =\frac{1}{4}\Bigl(1-\sqrt{17}+\sqrt{\bigl(1-\sqrt{17}\bigr)^2+16}\Bigr)$$
or, from this post,
$$\cos\frac{2\pi}{17}=\frac{-1+\sqrt{17}+\sqrt{34-2\sqrt{17}}+2\sqrt{17+3\sqrt{17}-\sqrt{34-2\sqrt{17}}-2\sqrt{34+2\sqrt{17}}}}{16}$$
II. For $p=257$:
See this post.
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