Calculating Cosine Sum: Solving Complex Equations for Even Roots

[Dinheiro]

Homework Statement

Calculate
$cos\frac{2\pi}{2n+1} + cos\frac{4\pi}{2n+1} + cos\frac{6\pi}{2n+1} + ... + cos\frac{2n\pi}{2n+1}$

Homework Equations

Complex equations, maybe :p

The Attempt at a Solution

Let's say
$z^{2n+1} = 1$
The sum is equivalent to the sum of the real even roots of the equation above. That's it. Ideas?

 

[Ray Vickson]

Homework Statement

Calculate
$cos\frac{2\pi}{2n+1} + cos\frac{4\pi}{2n+1} + cos\frac{6\pi}{2n+1} + ... + cos\frac{2n\pi}{2n+1}$

Homework Equations

Complex equations, maybe :p

The Attempt at a Solution

Let's say
$z^{2n+1} = 1$
The sum is equivalent to the sum of the real even roots of the equation above. That's it. Ideas?

What are YOUR ideas?

 

[Dinheiro]

Those were basically my ideas, I just didn't write down my complete attempt at the solution, sorry. But I could find out what was missing in my resolution xD Thanks, Ray

 

[vela]

How are we supposed to see what's missing if you don't show us what you did?

 

[HallsofIvy]

You mention that, in the complex plane, the given arguments lie equally spaced about the unit circle so did you consider using $cos(z)=(e^{iz}+ e^{-iz})/2$?

 

[Dinheiro]

Actually, I didn't solve it this way, but nice one though! Thanks, hallsoflvy