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Find all real polynomials $p(x)$ that satisfy $\sin( p(x) ) = p( \sin(x) )$ and prove they are indeed all.

Is there an easy way to prove this?

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Hint: We have \begin{align*} &\sin(p(x)) = p(\sin(x)),\tag{1}\\ &p'(x) \cos(p(x)) = p'(\sin(x)) \cos(x),\tag{2}\\ &|p'(2k\pi) \cos(p(2k\pi))| = |p'(0)|\tag{3} \end{align*} for every integer $k$. By $(1)$, $\sin(p(2k\pi)) = p(\sin(2k\pi)) = p(0)$. Hence $(3)$ gives $$ |p'(2k\pi)| \sqrt{1 - p(0)^2} = |p'(0)|.\tag{4} $$ Argue that $p(0)\neq\pm1$. Hence $|p'(2k\pi)|$ is a constant for every integer $k$. Infer that $p(x)$ is affine, i.e. $p(x)=ax+b$. Now, show that $b=0$ and $a\in\{-1,0,1\}$.

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Hint:

  • $p(\sin(x))$ is periodic.

I hope this helps ;-)

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