I have been given the homogeneous equation:
$$\xi (t) - \int_0^{\pi} \sin(x+t) \xi (x) dx = 0$$
and I have to show that the only solution is the trivial solution. This is the adjoint homogeneous equation of a Fredholm problem I have been given, and $\xi$ is thus a linear functional. In this case $\xi \in L^q$ ($1 \leq q < \infty$).
My initial thought was to expand $\sin(x+t)$ in the natural way, but this does not give me anything at first sight. I also considered writing it as a product of $L^2$-bases, but I am unsure what sequence (ansatz) to make.
Is there a trivial way to argue that the trivial solution must be the only solution?