Density of polynomials in $\cos t$ in $\mathcal{C}^0([0,\pi],\mathbb{R})$

Question

I'm looking at the Fourier cosine transform, and as a preliminary I have to show that every $f$ in $\mathcal{C}^0([0,\pi],\mathbb R)$ is the uniform limit of a sequence of functions of the form $t\to P(\cos t) $ where $P$ is a polynomial over the real numbers.

I have looked at showing that every polynomial function verifies the property, starting to the monomials, and conclude using the Stone-Weierstrass approximation theorem, but my attempt based on using a power series representation of $\arccos$ didn't work out.

Anyone got any hints ? Any help would be greatly appreciated.

Answer 1

An alternative to the ideas in the comments. The function $g(t)=f(\arccos t)$ is continuous in $[-1,1]$. $g$ is the uniform limit of a sequence of polynomials $p_n(t)$. Then $f(x)$ is the uniform limit of $p_n(\cos x)$.