Summation using Complex Numbers

Question

Prove that $$\sum_{k=0}^n \binom{n}{k} \cos((k+1)x) = 2^n \cos^n (x/2) \cos\left(\frac{n+2}{2}x\right)$$

I have no clue how to begin, so please provide me some hint to begin with

Answer 1

Hint: \begin{align} \sum_{k=0}^m\binom{m}{k}\cos(2k\theta) &=\Re\sum_{k=0}^m\binom{m}{k}e^{2ik\theta}\\ &=\Re(1+e^{2i\theta})^m \\ &=\Re\left( 2^m\left(\frac{e^{i\theta}+e^{-i\theta}}{2}\right)^me^{im\theta}\right)\\ &=2^m\cos^m\theta\cos m\theta \end{align}

Answer 2

Hint:

$$\cos kx=\frac12(e^{ikx}+e^{-ikx})$$