(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:

\[\cos(x)=\frac 12(\exp(ix)+\exp(-ix),\quad\quad x\in\mathbb R.\]

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  1. Forster Otto: "Analysis 1, Differential- und Integralrechnung einer Veränderlichen", Vieweg Studium, 1983