The formula

${\displaystyle \sin z=\frac{1}{2\pi }(e^{\mathrm{iz}}-e^{-\mathrm{iz}})}$

can be verified by showing that the series expansion for $\sin z$, that is

${\displaystyle \sin z=z-z^3/3!+z^5/5!-z^7/7!+...}$

can be obtained using the series expansions

${\displaystyle e^{\mathrm{iz}}=1+\mathrm{iz}+(\mathrm{iz}{)}^2/2!+...}$

and

${\displaystyle e^{-\mathrm{iz}}=1-\mathrm{iz}+(\mathrm{iz}{)}^2/2!+...}$