I am looking for any non-trivial solution to the following integral equation. That is, find any function $q(\theta)\neq 0$ which satisfies the following equation:
$$\int_0^a \sin(x\cos\theta)q(\theta)d\theta=0,$$ where $0<a<\pi/2$ and the equation holds for all $x\in\mathbb{R}$. Do non-trivial solutions even exist?
One can also consider performing a Jacobi-Anger expansion in which case a sufficient solution would be a $q(\theta)$ for which the following equation holds for all $n\in\mathbb{N}$:
$$\int_0^a \cos([2n-1]\theta)q(\theta)=0$$
If $a=2\pi$ then the solution would be simple ($q=\sin\theta$), but again $0<a<\pi/2$.
One thing to notice is that the kernel is non-singular. Does this tell us anything?
I tried the following, but it only leads to the trivial solution:
Taking the derivative with respect to $a$ gives $\sin(x\cos(a))q(a)=0$. But this only tells me that q must be zero at $a$.
Looking through the literature, I have noticed that essentially nobody considers this type of homogeneous integral equation. Why? Is it because it is extremely ill-posed? Or is the answer obvious?