Chapter 1 - Section 1.6 - Complex Numbers - 1.6 Exercises - Page 64: 45

$\dfrac{1}{1+i}-\dfrac{1}{1-i}=-i$

$\dfrac{1}{1+i}-\dfrac{1}{1-i}$ Evaluate the substraction: $\dfrac{1}{1+i}-\dfrac{1}{1-i}=\dfrac{(1-i)-(1+i)}{1-i^{2}}=\dfrac{1-i-1-i}{1-i^{2}}=\dfrac{-2i}{1-i^{2}}$ Substitute $i^{2}$ with $-1$: $...=\dfrac{-2i}{1-(-1)}=\dfrac{-2i}{2}=-i$