Let $f: [0,\infty) \rightarrow \mathbb{R}$ be a continuous function such that $f(0) = 0$. Is it true
that if the integral
$$
\int_0^{\pi/2} \sin(\theta) f(\lambda \sin(\theta)) \, d\theta
$$
is zero for every $\lambda > 0$, then $f$ is identically zero? 

I'm hoping this is true, and that"s perhaps why I'm stuck.