Euler's identity basically states that $\displaystyle e^{x\imath}=-1$ which is derived from $\displaystyle e^{x\imath}=\cos(x)+{\imath}\sin(x)$. (I'm using $\displaystyle \imath=$ i. Which I think is the correct use, though am unsure.)

I don't understand how they are getting to the latter form of the equation $\displaystyle e^{x\imath}=\cos(x)+{\imath}\sin(x)$. If someone could explain this to me it would be greatly appreciated.