Applying R425: Euler's formulas for a real variable, we have \begin{equation} \begin{split} e^{i \pi} + 1 & = \cos \pi + i \sin \pi + 1 \\ & = - 1 + i \cdot 0 + 1 \\ & = - 1 + 1 \\ & = 0 \end{split} \end{equation} $\square$