Questin on algebraic manipulation of exponential

Question

i don't understand how we went from the 2nd line to the 3rd line in order to have $\cos x$ and $\sin x$. This is used in order to prove that $\cos z$ is analytic, i know there is easier ways to achieve that but i want to understand this way.

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Answer 1

By Euler's formula $e^{ix}=\cos x+ i \sin x$, we have:

\begin{equation*} \frac{e^{-y}e^{ix}+e^ye^{-ix}}{2}= \frac{e^{-y}(\cos x+i \sin x)+e^y(\cos x- i \sin x )}{2} = \frac{(e^{-y}+e^y)\cos x + i (e^{-y}-e^y)\sin x}{2} \end{equation*}

Answer 2

For any real $x$, we have that:

$$e^{ix} = \cos(x) + i \sin(x).$$