Determine Fourier Series for $f(\theta) = e^{r\cos(\theta)}\cos(r\sin(\theta))$

Question

The book's suggestion is that $f(\theta) = Re(e^z)$ and it tells to expand $e^z$ using taylor series and show $f(\theta) = Re(e^z) = \sum_{n=0}^{+\infty} \frac{r^n}{n!}\cos(n\theta)$

but is it correct to assume $f(\theta) = Re(e^z)$