$$\frac{a^{-2}+b^{-2}}{a^{-1}+b^{-1}}·{\left(\frac{a^2+b^2}{ab}\right)}^{-1}$$

$$=\frac{{\displaystyle \frac{1}{a^2}}+{\displaystyle \frac{1}{b^2}}}{{\displaystyle \frac{1}{a}}+{\displaystyle \frac{1}{b}}}·{\left(\frac{ab}{a^2+b^2}\right)}^1$$

$$=\frac{{\displaystyle \frac{1}{a^2}}·{\displaystyle \frac{b^2}{b^2}}+{\displaystyle \frac{1}{b^2}}·{\displaystyle \frac{a^2}{a^2}}}{{\displaystyle \frac{1}{a}}·{\displaystyle \frac{b}{b}}+{\displaystyle \frac{1}{b}}·{\displaystyle \frac{a}{a}}}·\frac{ab}{a^2+b^2}$$

$$=\frac{{\displaystyle \frac{b^2}{a^2{b}^2}}+{\displaystyle \frac{a^2}{a^2{b}^2}}}{{\displaystyle \frac{b}{ab}}+{\displaystyle \frac{a}{ab}}}·\frac{ab}{a^2+b^2}$$

$$=\frac{{\displaystyle \frac{a^2+b^2}{a^2{b}^2}}}{{\displaystyle \frac{a+b}{ab}}}·\frac{ab}{a^2+b^2}$$

$$=\frac{\cancel{\left(a^2+b^2\right)}·ab}{\left(a^2{b}^2\right)·\left(a+b\right)}·\frac{ab}{\cancel{a^2+b^2}}$$

$$=\frac{ab·ab}{\left(a^2{b}^2\right)·\left(a+b\right)}$$

$$=\frac{\cancel{a^2{b}^2}}{\cancel{\left(a^2{b}^2\right)}·\left(a+b\right)}$$