Proof

By hypothesis, $x\in\mathbb R$ is a real number. It follows:
$\sin(-x)=\frac 1{2i}(\exp(i(-x))-\exp(-i(-x)),$ representing real sine by the complex exponential function,
$=\frac 1{2i}(\exp((-x) i)-\exp((-1)(-x)i),$ by commutativity of multiplying complex numbers,
$=\frac 1{2i}(\exp(-ix)-\exp(ix)),$ by rules of multiplying positive and negative real numbers,
$=\frac 1{2i}(-\exp(ix)+\exp(-ix)),$ by commutativity of adding complex numbers,
$=\frac 1{2i}((-1)(\exp(ix)-\exp(-ix))),$ by distributivity law for complex numbers,
$=- \frac 1{2i}(\exp(ix)-\exp(-ix))$
$=-\sin(x),$ once again applying representing of real sine by the complex exponential function.
Thus, the sine of a real variable is an odd function.
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Forster Otto: "Analysis 1, Differential- und Integralrechnung einer Veränderlichen", Vieweg Studium, 1983