TITLE: Is this (classical?) exercice missing a hypothesis?
QUESTION [9 upvotes]: A friend just told me about an exercice he was given quite a few years ago, but he wasn't sure wether he remembered all the hypothesis correctly. Does anybody recognize this?

Let $f$ be a smooth real-valued function on $\Bbb R$ such that for all $n\in\Bbb N,~\sup_{\Bbb R}|f^{(n)}|=1$ and $f(0)=1$, then $f=\cos$.

He wasn't sure wether theose were the exact hypothesis needed for this to work. He noticed that $f$ is automatically a power series with infinite radius of convergence, but wondered wether one ought to impose $f$ to be even, and wether the condition on the suprermums of the derivatives is correct.
NOTE. If you know the hypothesis, please post them as an answer, and if you want to post a solution, please hide the text so that one needs to scroll over it to reveal it, thanks!

REPLY [3 votes]: There is a 1980 paper by John Roe titled A characterization of the sine function, with the following abstract. I suspect the specific result here follows from Roe's theorem.

If a function and all its derivatives and integrals are absolutely
  uniformly bounded, then the function is a sine function with period
  2π.

http://journals.cambridge.org/action/displayAbstract?fromPage=online&aid=2081636