Inverting a pair of numbers

Suppose you have two numbers \(a \pm b\) both not equal to zero. Then
\[
(a \pm b)(a \mp b) = a^2 - b^2\\
\frac{1}{a \pm b} = \frac{a \mp b}{a^2 - b^2} = \frac{a}{a^2 - b^2} \pm \frac{-b}{a^2 - b^2}
\]