sequences and series - Find sum of $1 + cos theta + frac{1}{2!}cos 2theta + cdots$

Find the sum of following series:

$$1 + \cos \theta + \frac{1}{2!}\cos 2\theta + \cdots$$

where $\theta \in \mathbb R$.

My attempt: I need hint to start.

Answer

Hint:
$$1+\cos x + \frac{1}{2!}\cos 2x + \ldots = \Re(e^{0ix} + e^{1ix} + \frac{1}{2!}e^{2ix} + \ldots)=\Re e^{e^{ix}}$$

$$e^{ix}=\cos x+i\sin x\\\Longrightarrow e^{e^{ix}} = e^{\cos x}e^{i\sin x}=e^{\cos x}(\cos(\sin x)+i\sin(\sin x))$$
Your sum is

$$e^{\cos x}\cos(\sin x)$$