What is the Closed Form of a Summation of Sinusoidal Functions?

[Frillth]

Homework Statement

I am looking for a closed form of the summation:
sin(x) + sin(3x) + sin(5x) + ... + sin((2n-1*)x)

Homework Equations

None.

The Attempt at a Solution

Through a complete stroke of luck, I believe I have arrived at the correct solution: sin^2(nx)/sin(x)
I have tested this for many different cases, and I believe it is correct. However, I am having a hard time proving that it is. Can anybody point me in the right direction?

 

[rock.freak667]

How did you prove that?


[EDIT: The last term should be sin(2n+1)x not 2n-1

 

[Hurkyl]

Seems to me that mathematical induction would be an obvious thing to try...

 

[Frillth]

I'm a little rusty on my induction skills. Is this what I need to do?

1. Show that:
sin^2(nx)/sin(x) + sin((2(n+1)-1)x) = sin^2((n+1)x)/sin(x)

2. Show that my formula works for any specific case.

 

[rock.freak667]

You need to show that

$$\sum_{n=0} ^N sin(2n-1)x= \frac{sin^2(Nx)}{sinx}$$then add the (N+1)th term to each side and show that is can be written as

$$\frac{sin^2((N+1)x)}{sinx}$$

 

[Frillth]

I've been trying to get these two sides equal, but I'm not coming up with anything. Which identities should I be using to tackle this problem?