How can I prove the equality of the sum of cosines and sine?

Question

The task is to prove that for $k \in \mathbb{R} \setminus \{2 \pi z \; |\; z \in \mathbb{Z}\}$ $$ \frac{1}{2} +\sum_{i = 1}^{n} \cos(i k) = \frac{\sin((n + \frac{1}{2}) \cdot k)}{2 \sin(\frac{1}{2}k)} $$ I really have no idea left on how to prove that this equality is valid. Except for $$ \sin(x) = \frac{\exp(ix) - \exp(-ix)}{2i} \qquad\qquad \cos(x) = \frac{\exp(ix) + \exp(-ix)}{2} $$ I also know that $$ \tan(x) = \frac{\sin(x)}{\cos(x)} \qquad\text{and}\qquad \tan(x + y) = \frac{\tan(x) + \tan(y)}{1 - \tan(x)\tan(y)} $$ And of course the additional theorems for sine and cosine. I dont know if this might help me.. Can someone help me?