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 $f:\&gt;[0,\pi]\to{\mathbb R}$ to a continuous function $g:\&gt;[{-\pi},\pi]\to{\mathbb R}$ 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?
For (2)

take f(t) = Sin(t)

$\int_0^{\pi}$ sint sin(nt)dt = $\int_0^{\pi}$ (cos(n-1)t - cos(n+1)t) dt = $[\frac{sin(n-1)t}{n-1} - \frac{sin(n+1)t}{n+1}]_0^{\pi} = 0$ for all n $\in \mathbb N$

please tell me I am right or not

Thank you