How to write $\prod_{1\leq i<j \leq N}\exp \left(\alpha \cos (\theta_i - \theta_j)\right)$ in determinantal form?

Question

I came across this product in the context of a generalization of the usual probability distribution function in the Circular Unitary Ensemble (in Random Matrix Theory) with the original problem framed as: $$P(\theta_1, ..., \theta_N) \propto \prod_{1\leq i<j \leq N} |e^{i\theta_i} - e^{i\theta_j}|^2 \exp \left(\alpha \cos (\theta_i - \theta_j)\right) $$ for a constant $\alpha \in \mathbb{R}$. Obviously when $\alpha=0$, the problem reduces to the usual (as described by Mehta) CUE integrand and can be simplified in determinantal form to yield a characteristic kernel: $$\prod_{1\leq i<j \leq N} |e^{i\theta_i} - e^{i\theta_j}|^2 =\det \left[ \sum_{k=1}^N \frac{1}{2\pi} e^{i(k-\frac{N+1}{2})(\theta_i - \theta_j)}\right]\vert_{i,j=1}^N$$ Is there any special property from determinant theory that can be useful when $\alpha \neq 0$?