Evaluating Sum: \sum_{n=0}^N\frac{\cos{n\theta}}{\sin^n{\theta}}

[nicksauce]

Homework Statement

Evaluate the sum
$$\sum_{n=0}^N\frac{\cos{n\theta}}{\sin^n{\theta}}$$

Homework Equations

The Attempt at a Solution

In class we evaluated $$\sum_{n=0}^N\cos{n\theta}$$ and $$\sum_{n=0}^N\sin{n\theta}$$, by expanding them as the real and imaginary parts of a geometric series. However, I can't quite seem to figure out to use that for this question. Could someone give me a bump in the right direction?

 

[sarujin]

Maybe De Moivre's Theorem is useful here? Not sure if that's what you meant by expanding as real and imaginary parts.

[cos(theta) + i*sin(theta)]^n = cos(n*theta) + i*sin(n*theta)

 

[nicksauce]

For example, we used

$$\sum_{n=0}^{N}\cos{n\theta} = Re(\sum_{n=0}^{N}z^n)$$

And then use the analytic formula for the RHS.