Let $f :[0,\pi] \rightarrow \mathbb R$ be a continuous function such that $f(0) = 0$ . Which of the following statements are true? 1.If $$\int_0^{\pi} f(t)\cos(nt)~dt=0$$
for all $n\in\{0\}\cup\mathbb N$ , then $f\equiv0$ .
2.If $$\int_0^{\pi} f(t)\sin(nt)~dt=0$$
for all $n\in\{0\}\cup\mathbb N$ , then $f\equiv0$ .

A hint: Extend your function $f:\>[0,\pi]\to{\mathbb R}$ to a continuous function $g:\>[{-\pi},\pi]\to{\mathbb R}$ in two different ways: For 1. let $g$ be even, and for 2. let $g$ be odd.What can you say about the Fourier series of $g$ in the two cases?
