Sum of Cosines of Multiples of Angle

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

Proof
By the Simpson's Formula for Cosine by Sine:
 * $2 \cos \alpha \sin \beta = \map \sin {\alpha + \beta} - \map \sin {\alpha - \beta}$

Thus we establish the following sequence of identities:

Summing the above:


 * $\ds 2 \sin \frac x 2 \paren {\frac 1 2 + \sum_{k \mathop = 1}^n \map \cos {k x} } = \sin \frac {\paren {2 n + 1} x} 2$

as the sums on the form a telescoping series.

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 Sines of Multiples of Angle