Separating real and imaginary parts from a plane wave expansion

Question

From a plane wave equation $$e^{ix \cos{\varphi}}=\sum_{n=-\infty}^\infty J_{n}(x)[ie^{i\varphi}]^n$$ If I set $\varphi=0$, how can I separate them into real and imaginary parts? I want to know how the answer can become $$e^{ix}=\sum_{n=-\infty}^\infty i^n \space J_{n}(x)\tag1$$ and $$\cos(x)=J_0(x)+2\sum_{n=1}^\infty (-1)^n J_{2n}(x)\tag2$$