Mapping from Unit Circle defines Periodic Function

Theorem
Let $\SS$ denote the unit circle whose center is at the origin of the Cartesian plane $\R^2$.

Let $p: \R \to \SS$ be the mapping defined as:
 * $\forall x \in \R: \map p x = \tuple {\cos x, \sin x}$

Let $f': \SS \to \R$ be a real-valued function.

Then the composition $f' \circ p$ defines a periodic real function whose period is $2 \pi$.

Proof
Let $f := f' \circ p$ denote the composition of $f$ with $p$.

We have:

Hence the result.

Also see

 * Periodic Function as Mapping from Unit Circle