\( \left[\frac{1+i}{1-i}\right]^{r}+\frac{1}{x+i y}=1+i \)

Step 1 ：\( (1+i)^{r}(1-i)^{-r}+\frac{1}{x+iy} = 1+i \)

Step 2 ：Considering \( r = 1, x = \frac{1}{2}, y = \frac{1}{2} \)

Step 3 ：\( \frac{2i}{2} = 1+i \)