How to get the basis of $\cos(nx)$ in Chebyshev polynomials.

Question

Given $\cos(nx) = f_n(\cos x)$ for any positive integer $n$ and polynomial $f_n(x) = 2^{n-1}x^n + a_1 x^{n-1} + ... + a_{n-1}x + a_n$.

Show if $n$ is odd, $a_n = 0$ and if $n$ is even, $a_n=(-1)^{\frac n 2}$.

Answer 1

Consider $\cos\left(n{\pi\over2}\right)\ $.