$\begingroup$

Let $f\colon [0,\pi]\to\mathbb{R}$ be a continuous function. If $\int^{\pi}_{0}f (t) \sin(t)dt =\int^{\pi}_{0} f (t) \cos(t)dt = 0$, then $f(x)=0$ admits two solutions in $[0,\pi]$

I try to show if $f(x)>0$ and then get the contradiction but I failed to prove that, so maybe can someone help me with that? thanks in advance.