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?
 A: Well, we know Euler's formula:
$$e^{\theta i}=\cos\left(\theta\right)+\sin\left(\theta\right)i\tag1$$
So, we also get:
$$e^{\theta i}+e^{-\theta i}=2\cos\left(\theta\right)\space\Longleftrightarrow\space\cos\left(\theta\right)=\frac{e^{\theta i}+e^{-\theta i}}{2}\tag2$$
Now, we have:
$$2\cos\left(\frac{\pi}{5}\right)=\frac{1+\sqrt{5}}{2}=\phi=2\cdot\frac{e^{i\frac{\pi}{5}}+e^{-\frac{i\pi}{5}}}{2}=e^{i\frac{\pi}{5}}+e^{-\frac{i\pi}{5}}\tag3$$
A: 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. 

A: More generally, $\cos(nx) = T_n(\cos(x)$ where $T_n$ is the $n$'th Chebyshev polynomial of the first kind.  So, for example, since $T_5(t) = 16 t^5 - 20 t^3 + 5 t$, and $\cos(\pi) = -1$, we find that $t = \cos(\pi/5)$ satisfies
$$ 16 t^5 - 20 t^3 + 5 t + 1 = 0 $$
That polynomial can be factored as $(t+1)(4 t^2 - 2 t - 1)^2$, and since $t \ne -1$ we conclude that $t$ is a root of $4 t^2 - 2 t - 1$, thus $(1 \pm \sqrt{5})/4$.  
This method can be used to express the cosine of any rational multiple of $\pi$ as the root of some polynomial with rational coefficients.  However, in most cases there is not such a nice expression of this root.
