Sum of Sines of Multiples of Angle/Proof 2

Proof
Let $x$ be a real number that is not a integer multiple of $2 \pi$.

Let $k$ be a non-negative integer.

We have, from Euler's Formula:


 * $\map \exp {i k x} = i \sin k x + \cos k x$

Summing from $k = 0$ to $k = n$, we have:


 * $\displaystyle \sum_{k \mathop = 0}^n \map \exp {i k x} = i \sum_{k \mathop = 0}^n \sin k x + \sum_{k \mathop = 0}^n \cos k x$

As $\sin k x$ and $\cos k x$ are both real for real $k, x$, we have: