How would I divide
$\displaystyle 1/
1+i - 1/1-i$
Here is suggestion on a different way to approach this sort of problem.
Recall that $\displaystyle \dfrac{1}{z} = \dfrac{{\overline z }}{{\left| z \right|^2 }}$.
So $\displaystyle \dfrac{1}{{1 + i}} - \dfrac{1}{{1 - i}}$ becomes $\displaystyle \dfrac{{1 - i}}{2} - \dfrac{{1 + i}}{2}$ which is easier to deal with.