Sum of Sines of Multiples of Angle

Theorem
where $x$ is not an integer multiple of $2 \pi$.

Proof
By Simpson's Formula for Sine by Sine:
 * $2 \sin \alpha \sin \beta = \cos \left({\alpha - \beta}\right) - \cos \left({\alpha + \beta}\right)$

Thus we establish the following sequence of identities:

Summing the above:

The result follows by dividing both sides by $2 \sin \dfrac x 2$.

It is noted that when $x$ is a multiple of $2 \pi$ then:
 * $\sin \dfrac x 2 = 0$

leaving the undefined.

Also see

 * Sum of Cosines of Multiples of Angle