Let $ f $ be a continuous function defined on $ [0,\pi] $. Suppose that
$$ \int_{0}^{\pi}f(x)\sin {x} dx=0, \int_{0}^{\pi}f(x)\cos {x} dx=0 $$
Prove that $ f(x) $ has at least two real roots in $ (0,\pi) $
Let $ f $ be a continuous function defined on $ [0,\pi] $. Suppose that
$$ \int_{0}^{\pi}f(x)\sin {x} dx=0, \int_{0}^{\pi}f(x)\cos {x} dx=0 $$
Prove that $ f(x) $ has at least two real roots in $ (0,\pi) $