Deriving Bessel generating function

So I am trying to prove that

$\displaystyle exp\left[\frac{x}{2}(t-\frac{1}{t}\right]=\sum^{\infty}_{n=-\infty} J_{n}(x)t^{n}$

What I did so far is to set $\displaystyle t=e^{i \theta}$ and I came down to something like that using fourier series

$\displaystyle exp\left[\frac{x}{2}(t-\frac{1}{t}\right]=\frac{2i\sin(\pi\sin \theta)}{\pi}\left[\frac{1}{2i\sin \theta} + \sum^{\infty}_{n=1} \frac{(-1)^{n}}{n^{2}-\sin^{2} \theta} (i\sin\theta\cos nx - n\sin nx)\right]$

I am not even sure it is correct or if it is the right way, I am just confused as to where to go from here. Any suggestions??? I really need help this thing is due soon. Any help is welcome!!!