# How do you simplify the cosine of this special angle?

The cosine value of 2π/17 is found to be the following result.

ToRadicals[Cos[(2 \[Pi])/17]]


get the result:

1/(4 Sqrt[2/(
15 + Sqrt[17] - Sqrt[2 (17 - Sqrt[17])] + Sqrt[
2 (34 + 6 Sqrt[17] + Sqrt[2 (17 - Sqrt[17])] - Sqrt[
34 (17 - Sqrt[17])] + 8 Sqrt[2 (17 + Sqrt[17])])])])


How to perform the operations to obtain this value shown in the image below?

poly = Factor[MinimalPolynomial[Cos[2 Pi/17], x], Extension -> {Sqrt[17]}]
x /. Solve[poly == 0 && x == Cos[2 Pi/17], x][[1]] // ToRadicals // FullSimplify


$$-\left(\left(1+\left(-4-2 \sqrt{17}\right) x+\left(6+2 \sqrt{17}\right) x^2+\left(-4+4 \sqrt{17}\right) x^3-16 x^4\right) \left(-1+\left(4-2 \sqrt{17}\right) x+\left(-6+2 \sqrt{17}\right) x^2+\left(4+4 \sqrt{17}\right) x^3+16 x^4\right)\right)$$

$$\frac{1}{16} \left(-1+\sqrt{17}+\sqrt{34-2 \sqrt{17}}+\sqrt{68+12 \sqrt{17}-4 \sqrt{170+38 \sqrt{17}}}\right)$$

or

ResourceFunction["Cos2PiOverFermatPrime"][17, Cos] // FullSimplify


$$\frac{1}{16} \left(-1+\sqrt{17}+\sqrt{34-2 \sqrt{17}}+\sqrt{68+12 \sqrt{17}-4 \sqrt{170+38 \sqrt{17}}}\right)$$