Evaluate $ \int_{-\infty}^{\infty} \frac{\exp(iwx)}{x^2-2x\cos(\theta) +1} dx $

Question

Using contour integration I need to evaluate the following integral. I presume there is a clever way of rewriting the function in terms of z.

Let $\theta \in (0,\pi) $ and $ w > 0$

$$ \int_{-\infty}^{\infty} \frac{\exp(iwx)}{x^2-2x\cos(\theta) +1} dx $$

Thanks in advance!