Properties of random variables with $E[ \sin(X+Z) \sin(3X+Z)]=0. $

Question

Let $X$ and $Z$ be two independent symmetric random variables such that \begin{align} E[ \sin(X+Z) \sin(3X+Z)]=0. \end{align}

From the above identity ca we say something about the distributions of $X$ and $Z$? Or is it hopeless? Thanks.