Let $C[0,\pi]$ denote the vector space of all continuous functions on the closed interval $[0,\pi]$. Does there exist a non-zero function $x(t)\in C[0,\pi]$ such that the equation $$\int_0^\pi (\sin(s)+\cos(t))x(t)\;dt=0$$ holds for all $s\in[0,\pi]$?
I think that a function $x(t)\in C[0,\pi]$ with $x(\pi)=0$ does the job, but I could not find an explicit function $x(t)$.
Thanks in advance for a hint or a solution.
An easy example: $x(t) = \cos(2t)$.
More generally: every function $x(t)$ which satisfies both
$$\int_0^\pi \cos(t) x(t) dt = 0$$
and
$$\int_0^\pi x(t) dt = 0$$
will work, and it is easy to see that in fact all the functions
$$x_n(t) = \cos(2nt) \quad (n \in \mathbb{N})$$
work as they satisfy these requirements.