Answer
$-i$
Work Step by Step
$\displaystyle \frac{1}{1+i}-\frac{1}{1-i}=\frac{1}{1+i}*\frac{1-i}{1-i}-\frac{1}{1-i}*\frac{1+i}{1+i}=\frac{1-i}{1-i^{2}}-\frac{1+i}{1-i^{2}}=\frac{1-i}{1--1}-\frac{1+i}{1--1}=\frac{1-i}{2}+\frac{-1-i}{2}=\frac{-2i}{2}=-i$