Prove that $\displaystyle \int_0^{2\pi}e^{\cos\theta}\cos(\sin\theta)=2\pi$
Differentiate under the integral sign, it'll be helpful to get what you want.
Define $\displaystyle f(\varphi ) = \int_0^{2\pi } {e^{\varphi \cos \theta } \cos (\varphi \sin \theta )} \,d\theta.$
From there, you just need to find $\displaystyle f(1),$ and it's done.
(You forgot the $\displaystyle d\theta.$)