Question Number 48225 by gunawan last updated on 21/Nov/18
$$\mathrm{prove}\:\mathrm{that} \\ $$$$\mathrm{exp}\left(\frac{\mathrm{2}+\pi\mathrm{i}}{\mathrm{4}}\right)=\sqrt{\frac{{e}}{\mathrm{2}}}\left(\mathrm{1}+{i}\right) \\ $$$$\mathrm{cos}\:\left({z}_{\mathrm{1}} +{z}_{\mathrm{2}} \right)=\mathrm{cos}\:{z}_{\mathrm{1}} \mathrm{cos}\:{z}_{\mathrm{2}} −\mathrm{sin}\:{z}_{\mathrm{1}} \mathrm{sin}\:{z}_{\mathrm{2}} \\ $$
Answered by Smail last updated on 21/Nov/18
$${e}^{\frac{\mathrm{2}+\pi{i}}{\mathrm{4}}} ={e}^{\frac{\mathrm{1}}{\mathrm{2}}+{i}\frac{\pi}{\mathrm{4}}} =\sqrt{{e}}×{e}^{{i}\pi/\mathrm{4}} =\sqrt{{e}}×\left({cos}\left(\frac{\pi}{\mathrm{4}}\right)+{isin}\left(\frac{\pi}{\mathrm{4}}\right)\right) \\ $$$$=\sqrt{{e}}\left(\frac{\sqrt{\mathrm{2}}}{\mathrm{2}}+{i}\frac{\sqrt{\mathrm{2}}}{\mathrm{2}}\right)=\frac{\sqrt{{e}}}{\sqrt{\mathrm{2}}}\left(\mathrm{1}+{i}\right) \\ $$$$=\sqrt{\frac{{e}}{\mathrm{2}}}\left(\mathrm{1}+{i}\right) \\ $$