Binomial Summation

Question

The sum $$ 1 + {n \choose 1}\cos \theta + {n \choose 2}\cos 2\theta + \cdots+ {n \choose n}\cos n\theta $$ is?

I try to write this as the real part of $(1 + \cos \theta + i\sin \theta)^n$ but then I'm stuck.

Answer 1

The given sum is the real part of

$$\sum_{k=0}^n{n\choose k}e^{ik\theta}=(1+e^{i\theta})^n=e^{in\theta/2}\left(2\cos\left(\frac{\theta}2\right)\right)^n$$ so the desired sum is $$2^n\cos^{n}\left(\frac{\theta}2\right)\cos\left(\frac{n\theta}2\right)$$