${\left(\frac{1+\cos {\displaystyle \frac{\mathrm{\pi }}{8}}-\mathrm{i} \sin {\displaystyle \frac{\mathrm{\pi }}{8}}}{1+\cos \frac{\mathrm{\pi }}{8}+\mathrm{i} \sin \frac{\mathrm{\pi }}{8}}\right)}^8=$

$\begin{array}{rcl} & & {\left(\frac{1+\cos {\displaystyle \frac{\mathrm{\pi }}{8}}-\mathrm{i} \sin {\displaystyle \frac{\mathrm{\pi }}{8}}}{1+\cos \frac{\mathrm{\pi }}{8}+\mathrm{i} \sin \frac{\mathrm{\pi }}{8}}\right)}^8\end{array}$

$\begin{array}{rcl} & =& {\left(\frac{2{\cos }^2{\displaystyle \frac{\mathrm{\pi }}{16}}-\mathrm{i}\left(2\sin {\displaystyle \frac{\mathrm{\pi }}{16}} \cos {\displaystyle \frac{\mathrm{\pi }}{16}}\right)}{2{\cos }^2{\displaystyle \frac{\mathrm{\pi }}{16}}+\mathrm{i} \left(2\sin {\displaystyle \frac{\mathrm{\pi }}{16}} \cos {\displaystyle \frac{\mathrm{\pi }}{16}}\right)}\right)}^8\end{array}$

$\begin{array}{rcl} & =& {\left(\frac{2\cos {\displaystyle \frac{\mathrm{\pi }}{16}}\left(\cos {\displaystyle \frac{\mathrm{\pi }}{16}}-\mathrm{i} \sin {\displaystyle \frac{\mathrm{\pi }}{16}}\right)}{2\cos {\displaystyle \frac{\mathrm{\pi }}{16}}\left(\cos {\displaystyle \frac{\mathrm{\pi }}{16}}+\mathrm{i} \sin {\displaystyle \frac{\mathrm{\pi }}{16}}\right)}\right)}^8\end{array}$

$\begin{array}{rcl} & =& {\left(\cos \frac{\mathrm{\pi }}{16}-\mathrm{i} \sin \frac{\mathrm{\pi }}{16}\right)}^{16}\end{array}$

$\begin{array}{rcl} & & \\ & =& \cos \mathrm{\pi }-\mathrm{i} \sin \mathrm{\pi }\\ & =& -1\end{array}$