Can the exact value of cosine function be expressed as some finite combination of integers. nth power and fundamental operations($+,-,/,\times)$. When $x$ belongs to integers (in degrees) and is not a multiple of $3$. For example $\cos36=\frac{1+\sqrt 5}{4}$.
I tried to find the value of $\cos 1$ using a cubic equation but I ended up with a Casus irreducibilis.
Suppose that for some integer $k$ not divisible by $3$, the cosine of the $k$-degree angle can be expressed using arithmetical operations, including square root.
Note first that we can express the cosine of $3^{\circ}$ using the operations of arithmetic and square root. For as pointed out by the OP, the cosine of $36^\circ$ is so expressible. The cosine of the $30^\circ$ angle is also so expressible, so by using the cosine of a difference formula, the cosine of $6^\circ$ is so expressible, and therefore so is the cosine of $3^\circ$.
There are integers $s$ and $t$ such that $3s+kt=1$. So if the cosine of the $k$-degree angle is expressible, so the cosine of the $1$-degree angle, and hence the cosine of the $20$ degree angle. However, it is a well-known theorem that the cosine of the $20$ degree angle cannot be so expressed.
The result can be strengthened. The cosine of $20^\circ$ cannot be expressed using the operations of arithmetic, and $n$-th roots, where we are not allowed to take the $n$-th root of a number that is not real. The same result therefore holds for the cosine of $k$-degree angles, where $k$ is an integer not divisible by $3$.
