Is there a naive proof that $x - \frac{x^3}{3!} + \frac{x^5}{5!} - \ldots$ has period $2\pi$?
There is a companion to that series,
$$g(x)=1-\frac{x^2}2+\frac{x^4}{4!}+\cdots$$
and together they have a funny property: $$h(x+y):=g(x+y)+if(x+y)=(g(x)+if(x))(g(y)+if(y))=h(x)h(y).$$
This can be shown in an elementary way by developing the powers of $x+y$ using the binomial theorem and identifying to the products of the partial sums.
Then assuming that by some magic (such as the intermediate value theorem) we can show that there is a solution to
$$h(2\pi)=1,$$
where $\pi$ is the unknown, then for all $x$
$$h(x+2\pi)=h(x).$$