Asymptotic expansion of the series $\sum_{n=-\infty}^{\infty} \frac{\sin(\theta)}{-\cosh[(2n+1)\pi x] + \cos(\theta)}$

Question

I would like to find the asymptotic expansion of

$$ I(x)= \sum_{n=-\infty}^{\infty} \frac{\sin(\theta)}{-\cosh[(2n+1)\pi x] + \cos(\theta)} $$ when $x \rightarrow 0$. Here $-\pi < \theta < \pi$. I think that the solution is $$ I(x) \sim \frac{1}{x}( \frac{\theta}{\pi} -1 ) \ (x \rightarrow 0)$$ but I don't know how to prove it.