(related to Corollary: Representing Real Cosine by Complex Exponential Function)
By definition of the cosine function of a real variable, we have \[\cos(x)=\Re(\exp(ix)),\quad\quad x\in\mathbb R.\]
From the formula of extracting the real part from a complex number, we get the result
\[\cos(x)=\Re(\exp(ix))=\frac 12(\exp(ix)+(\exp(ix))^*),\quad\quad x\in\mathbb R.\]
By the formula for calculating the complex conjugate of the complex exponential function, it follows
\[\cos(x)=\frac 12(\exp(ix)+\exp((ix)^*),\quad\quad x\in\mathbb R.\]
The definition of the complex conjugate leads to the desired result:
\[\cos(x)=\frac 12(\exp(ix)+\exp(-ix),\quad\quad x\in\mathbb R.\]
