Simplify a series of $\cos x$ or $\sin x$ Suppose we have this series 
$$\sum\limits_{k=0}^n a^k \cos(kx) = 1 + a\cos(x) + a^2 \cos(2x) + a^3 \cos(3x) + \cdots + a^n\cos(nx)$$
What should we do to simplify this?

You should give more information about what you have tried. In the meantime 
Hint: $$\cos kx =\frac {e^{ikx}+e^{-ikx}}2$$
