(related to Corollary: Representing Real Sine by Complex Exponential Function)

By definition of the sine function of a real variable, we have \[\sin(x)=\Im(\exp(ix)),\quad\quad x\in\mathbb R.\]

From the formula of extracting the real part from a complex number, we get the result

\[\sin(x)=\Im(\exp(ix))=\frac 1{2i}(\exp(ix)-(\exp(ix))^*),\quad\quad x\in\mathbb R.\]

By the formula for calculating the complex conjugate of the complex exponential function, it follows

\[\sin(x)=\frac 1{2i}(\exp(ix)-\exp((ix)^*),\quad\quad x\in\mathbb R.\]

The definition of the complex conjugate leads to the desired result:

\[\sin(x)=\frac 1{2i}(\exp(ix)-\exp(-ix)),\quad\quad x\in\mathbb R.\]

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