Inverting a pair of numbers

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