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How to prove the periodicity of an iterated function?

For example, how to prove $\sin_{[n]}(x)$ is a periodic function of period $2\pi$ $\forall n\in\mathbb{R}^+\cap\{0\}$ ?

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In total generality, if $f: \mathbb{R} \rightarrow \mathbb{R}$ is periodic with period $p$, i.e. $f(x+p) = f(x)$ for all $x$, then on the nose $g \circ f$ has period $p$ for any $g$.

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