integration - How do I prove $int^{pi}_{0} frac{cos nx}{1+cosalpha cos x} mathrm{d}x = frac{pi}{sin alpha} (tan alpha - sec alpha)^n$?

The result I wish to show is that for $n \in \mathbb{Z}$, $$\int^{\pi}_{0} \frac{\cos nx}{1+\cos\alpha \cos x} \mathrm{d}x = \frac{\pi}{\sin \alpha} (\tan \alpha - \sec \alpha)^n$$

I have made a few attempts through the first techniques that came to my mind but I have not made any meaningful progress.