Polynomial equation involving $\cos(x)$

Question

After some chitchat I stumbled upon the following problem

What are the real valued-polynomials $p$ verifying $$\cos(p(x))=p(\cos(x))$$

My findings :

, $p(x)=x$ works

But besides this one, I cannot find any, any clues ?

Thanks for the help,

T.D

Answer 1

$$p(\cos(x))$$ is a periodic function of period $2\pi$. But $$\cos(p(x))$$ is aperiodic unless $p$ is a linear function of $x$.

Now

$$a\cos(x)+b=\cos(ax+b)$$

is only possible with $a=1,b=0$ because the LHS alternates between $a+b$ and $a-b$, which must match $1,-1$.

Anyway, another option exists, when $a=0$ (so that the RHS does not alternate), with

$$b=\cos(b),$$ which has a single real solution $b=0.7390851332152\cdots$.