Exact value for the converging sum of the product of a geometric series and cosine function

Question

$\sum\limits_{i=0}^\infty (\frac{1}{C})^n \dot \cos(\frac{(n+1)\pi}{a}) $

I was working with this sum and found that it converged (when C > 1) but was wondering how to find the exact value. By analyzing a few cases through WolframAlpha I found that the exact value was represented by the equation below.

$ \frac{C^2\sin(\frac{\pi}{a})}{C^2+1-2C\cos(\frac{\pi}{a})} $

I was hoping someone could point me towards deriving this answer, it would be greatly appreciated.

Answer 1

$u^n\cos\dfrac{(n+1)\pi}a$ is the real part of $$u^ne^{i(n+1)\pi/a}=e^{i\pi/a}\cdot(u e^{i\pi/a})^n$$ which is the $n$th term of Geometric Series with the first term $=e^{i\pi/a}$ and common ratio $=u e^{i\pi/a}$

Check the proof of Convergence