I'm failing to understand the following transition:
Given that $f$ is an odd function defined on $[-\pi,\pi]$, we know that for all $\epsilon>0$ there exist $a_n, b_n \in \mathbb{R}$ such that:
$$||f-\sum_{n=0}^Na_n \sin(nx)+b_n\cos(nx)||_\infty<\epsilon$$
(where this is the sup norm, i.e. max value in $[-\pi,\pi]$ in this case),
then for all $\epsilon$ there exist $a_n$ such that:
$$\max_{[0,\pi]} \big|f-\sum_{n=0}^N a_n \sin(nx)\big|<\epsilon$$
This transition is supposed to rely on the fact that both $f$ and $\sin$ are odd functions somehow, but I just can't see how to prove this (this should be simple, from my gathering, so it can't be a matter of being familiar with complicated trigonometric identities etc.).
Any help would be greatly appreciated.