Use Demoivre's theorem. I guess you write
i = cos π/2 + i sin π/2 = eiπ/2 => √i = eiπ/4 => cos π/4 + i sin π/4
-i = cos -π/2 + i sin -π/2 = e-iπ/2 => √-i = e-iπ/4 => cos -π/4 + i sin -π/4
Add both.
Use Demoivre's theorem. I guess you write
i = cos π/2 + i sin π/2 = eiπ/2 => √i = eiπ/4 => cos π/4 + i sin π/4
-i = cos -π/2 + i sin -π/2 = e-iπ/2 => √-i = e-iπ/4 => cos -π/4 + i sin -π/4
Add both.