How do I evaluate $\cos(x) + \cos (2x) +\cos (3x) + ... + \cos (nx)$?

Question

How to evaluate the above expression and express the answer in terms of $n$ and $x$?

Answer 1

Hint

The desired sum is the real part of the geometric sum:

$$\sum_{k=1}^n e^{ikx}$$

Answer 2

hint multiply numerator and denominator by $\sin(x/2)$ and then use a telescopic series.