Let $(f,g)$ be a pair of real-valued $C^1$ functions on $\mathbb{R}$ satisfying
$$\forall x\in\mathbb{R}\left(f'(x)=g(x)\quad\text{and}\quad g'(x)=-f(x)\right)$$
$$f(0)=0\quad\text{and}\quad g(0)=1$$
Then it is pretty immediate that $f$ and $g$ are both $C^\infty$ on $\mathbb{R}$, this determines a power series which determines uniqueness of the pair. A cute argument (without power series) shows that these functions satisfy $f(x)^2+g(x)^2=1$.

Here's my question: how do we deduce that $f(x)$ and $g(x)$ are $2\pi$ periodic? I understand that we could have started the entire axiomatic system by defining $f(x)$ and $g(x)$ in terms of triangles, but since we already have uniqueness from this setup I wonder how we deduce periodicity from here.