Q:

$\cos{nx}=p(\cos{x})$, general formula for $p$?

Does a general, closed formula for the $p$ polynom exist?
$\cos{nx}=p(\cos{x})$

A:

Use the Chebyshev polynomials: From $T_n(y) = \cos(n \arccos(y))\;$ you get with $x=\arccos(y):$
$$\cos(nx) = T_n(\cos(x))$$
The $T_n$ can be generated with the standard formulas for these orthogonal polynomials:
$$T_0(x) = 1$$
$$T_1(x) = x$$
$$T_{n+1}(x) = 2x\,T_{n}(x) - T_{n-1}(x)$$

