sequences and series - Evaluating $1+frac{cos(theta)}{1!}+frac{cos(2theta)}{2!}+frac{cos(3theta)}{3!}+cdots$

Problem: We are asked to evaluate the infinite sum
$$1+\frac{\cos(\theta)}{1!}+\frac{\cos(2\theta)}{2!}+\frac{\cos(3\theta)}{3!}+\cdots$$

Thoughts on the problem: At first the infinite sum looks very similar to the power series expansion of the exponential function. However, instead of $\cos(\theta)^n$, we have $\cos(n\theta)$. I tried to use trig identities for $\cos(n\theta)$, but nothing seemed to jump out at first.

I know that $\cos(n\theta)$ is the $n$th Chebyshev polynomial evaluated at $\cos(\theta)$, and I was hoping that would allow us to simplify the expression somewhat. However, I do not know of any identities relating sums of these polynomials, so this did not get me very far.

Can anyone share a useful way to use the Chebyshev polynomials or give a reason why we should not use them? If not, what would be another good starting point? Thanks.

Answer

Hint

$$\cos(n\theta)+i\sin(n\theta)=e^{in\theta}$$