Question Number 138362 by mohammad17 last updated on 12/Apr/21
$${prove}\:{that}\: \\ $$$$ \\ $$$$\left(\mathrm{1}+{i}\right)^{{n}} +\left(\mathrm{1}−{i}\right)^{{n}} =\mathrm{2}^{\frac{{n}+\mathrm{1}}{\mathrm{2}}} \\ $$
Commented by Dwaipayan Shikari last updated on 12/Apr/21
$$\left(\mathrm{1}+{i}\right)^{{n}} +\left(\mathrm{1}−{i}\right)^{{n}} \\ $$$$=\mathrm{2}^{\frac{{n}}{\mathrm{2}}} {e}^{\frac{\pi{n}}{\mathrm{4}}{i}} +\mathrm{2}^{\frac{{n}}{\mathrm{2}}} {e}^{\frac{−\pi{n}}{\mathrm{4}}{i}} \\ $$$$=\mathrm{2}^{\frac{{n}}{\mathrm{2}}+\mathrm{1}} {cos}\left(\frac{\pi{n}}{\mathrm{4}}\right) \\ $$