Odd Cosn Problem: What is cosπ/n+cos3π/n+...+cos(2n-1)π/n? Prove It!

[xuying1209]

if n is an odd, cosπ/n+cos3π/n+cos5π/n+...+cos(2n-1)π/n is equal to what?
And how can I prove it??

 

[HallsofIvy]

For n= 1, cos(pi/1)= -1.

For n> 1, n odd, essentially you are adding the real parts of the 2nth roots of unity. Since those roots are symmetric about the imaginary axis, the sum is 0.

 

[tehno]

Ahh that it was... almost racked my brains out 'cause "π" I read as n ( not $\pi$) ...

 

[robert Ihnot]

I am not sure what is being asked. Is this $$\sum cos(n_i)/n_i, or \sum cos(pi*n_i/n_i), or what?$$

 

[HallsofIvy]

I interpreted as sum of [itex] cos(i\pi/n)[/tex] for i= 1 to n-1.