Ersetze in der Formel ${\displaystyle \tan z={\frac {1}{i}}\,{\frac {e^{iz}-e^{-iz}}{e^{iz}+e^{-iz}}}\quad z}$ durch ${\displaystyle x_{1}+...+x_{n}\,}$

${\displaystyle \tan(x_{1}+...+x_{n})={\frac {1}{i}}\,{\frac {e^{i(x_{1}+...+x_{n})}-e^{-i(x_{1}+...+x_{n})}}{e^{i(x_{1}+...+x_{n})}+e^{-i(x_{1}+...+x_{n})}}}}$

Schreibe ${\displaystyle e^{\pm i(x_{1}+...+x_{k})}}$ als ${\displaystyle e^{\pm ix_{1}}\cdots e^{\pm ix_{n}}}$ und jeden Faktor ${\displaystyle e^{\pm ix_{k}}}$ als ${\displaystyle \cos x_{k}\pm i\sin x_{k}\,}$.

${\displaystyle \tan(x_{1}+...+x_{n})={\frac {1}{i}}\,{\frac {\prod _{k=1}^{n}\left(\cos x_{k}+i\sin x_{k}\right)-\prod _{k=1}^{n}\left(\cos x_{k}-i\sin x_{k}\right)}{\prod _{k=1}^{n}\left(\cos x_{k}+i\sin x_{k}\right)+\prod _{k=1}^{n}\left(\cos x_{k}-i\sin x_{k}\right)}}}$

Teile nun Zähler und Nenner jeweils durch ${\displaystyle \prod _{k=1}^{n}\cos x_{k}}$.