2D-Fourier transform of $(r,\theta) \mapsto (1 + \mathrm{i} \cos ( \theta ))^{-1}$?

Question

Consider the function $f : \mathbb{R}^2 \rightarrow \mathbb{R}$ defined in polar coordinates by $$ f(r,\theta) = \frac{1}{1 + \mathrm{i} \cos ( \theta )}.$$ The function $f$ is in $\mathcal{S}'(\mathbb{R^2})$ so that its Fourier transform is well-defined.

Question: Is there a nice closed-form expression for the Fourier transform of $f$?