Complex representation of a real wave

Question

I have the following formula for a wave: $u(x)=a \cos(kx-\omega t + \phi)$. I am trying to recover the complex representation of it $u(x)=\Re(a e^{i\phi}e^{i(kx-wt)}$. I started by using Euler's formula $e^{i\theta}=\cos\theta+i\sin\theta$ to write $\cos\theta=\frac{e^{i\theta}+e^{-i\theta}}{2}$ and plugging it in the original $u(x)$ formula. I do not know how to proceed further from here, as there is a sum and not a product of exponentials. If I could just randomly change the sign of $e^{-i\theta}$ I would be done, but I do not know any rule that allows this...