Can the cosine function be written in "simpler" expressions? I recently found that the constants $\pi$ and $\phi=\frac{1+\sqrt{5}}{2} $ can be related by the identity 
$$
2\cos \frac{\pi}{5} = \phi.
$$
Is there some way to write the cosine function by "simpler" mathematical expressions?

Gauss introduced a method for cyclotomic polynomials that quickly gives such expressions. Let $n$ be odd and $\omega$ a primitive $n$-th root of unity. Then one of th values indicated by the expression $\omega + (1/\omega)$ is, indeed, $ x =2 \cos (2 \pi / n).$ These satisfy monic polynomials with integer coefficients. If you prefer $ x =2 \cos ( \pi / n)$ you need half angle formulas.
Oh, given the way these are constructed, all the roots are real and given by $2 \cos 2k\pi/n$
Fairly consistent for prime $n.$
$n=5$
$$ x^2 + x - 1 $$
$n=7$
$$ x^3 + x^2 -2x-1 $$
$n=11$
$$ x^5 + x^4 - 4 x^3 - 3 x^2 + 3x +1 $$
$n=13$
$$  x^6 + x^5 - 5 x^4 - 4 x^3 + 6 x^2 + 3 x - 1 $$
$n=17$
$$ x^8 + x^7 - 7 x^6 - 6 x^5 + 15 x^4 + 10 x^3 - 10 x^2 - 4 x + 1  $$
$n=19$
$$ x^9 + x^8 -8 x^7 - 7 x^6 +21 x^5 + 15 x^4 -20 x^3 - 10 x^2 +5 x + 1  $$
These are all in pages 3-23 of Reuschle (1875). I learned the method in Galois Theory by David A. Cox, and wrote working programs for fixed polynomial degrees 3, 5, 7. 

