Computing fourier coefficients

Question

Im given that $J_n(x)$ are the fourier coefficients of $e^{ix\cdot\sin t} = \sum_{-\infty}^\infty J_n(x)e^{int}.$ I want to show that the sum $\sum_{-\infty}^\infty |J_n(x)|^2$ is not dependent on the variable $x$. I was thinking that the since the $J_{n}(x)$ are the coefficients, we can compute them: $J_n(x) = \frac 1 {2\pi} \int_{-\pi}^\pi e^{ix\sin t} e^{-int} \, dt.$ But this integral equals to $0$, so Im not sure what I should do?