$${\left(\frac{1+i}{\sqrt{2}}\right)}^{1000}+{\left(\frac{1-i}{\sqrt{2}}\right)}^{1000}$$

$$={\left[{\left(\frac{1+i}{\sqrt{2}}\right)}^2\right]}^{500}+{\left[{\left(\frac{1-i}{\sqrt{2}}\right)}^2\right]}^{500}$$

$$={\left(\frac{1+2i+i^2}{2}\right)}^{500}+{\left(\frac{1-2i+i^2}{2}\right)}^{500}$$

$$={\left(\frac{\cancel{1}+2i\cancel{-1}}{2}\right)}^{500}+{\left(\frac{\bcancel{1}-2i\bcancel{-1}}{2}\right)}^{500}$$

$$={\left(\frac{2i}{2}\right)}^{500}+{\left(\frac{-2i}{2}\right)}^{500}$$

$$=i^{500}+(-i{)}^{500}$$

$$=i^{500}+(-1{)}^{500}·i^{500}$$

$$=i^{4·125}+i^{4·125}$$

$$=1+1$$