$\text{If }i=\sqrt{-1}\text{, then }(1+i{)}^{10}+(1-i{)}^{10}=$

$=(1+i{)}^{10}+(1-i{)}^{10}$

$=(\sqrt{2}\left(\frac{1+i}{\sqrt{2}}\right){)}^{10}+(\sqrt{2}\left(\frac{1-i}{\sqrt{2}}\right){)}^{10}$

$=2^5{\left(\frac{1+i}{\sqrt{2}}\right)}^{10}+2^5{\left(\frac{1-i}{\sqrt{2}}\right)}^{10}$

$=2^5{\left(\cos \frac{\mathrm{\pi }}{4}+i\sin \frac{\mathrm{\pi }}{4}\right)}^{10}+2^5{\left(\cos \frac{\mathrm{\pi }}{4}-i\sin \frac{\mathrm{\pi }}{4}\right)}^{10}$

$=2^5\left(\cos \frac{5\mathrm{\pi }}{2}+\cancel{\mathrm{isin}\frac{5\mathrm{\pi }}{2}}+\cos \frac{5\mathrm{\pi }}{2}-\cancel{i\sin \frac{5\mathrm{\pi }}{2}}\right)$

$=0$