\[\left( a^{- 1} + b^{- 1} \right)(a + b)^{- 1} =\]
\[= \left( \frac{1^{\backslash b}}{a} + \frac{1^{\backslash a}}{b} \right) \cdot \frac{1}{a + b} =\]
\[= \frac{b + a}{\text{ab}} \cdot \frac{1}{a + b} = \frac{1}{\text{ab}}.\]