If the expression $(x + y)^{- 1} . (x^{- 1} + y^{- 1}) (x y^{- 1} + x^{- 1} y)^{- 1}$ is simplified, it takes the form of which one of the following?

x + y

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(x^{2} + y^{2})^{- 1}

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x^{2} + y^{2}

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Solution

The correct option is D $(x^{2} + y^{2})^{- 1}$
Given expression
$= \frac{1}{x + y} \cdot (\frac{1}{x} + \frac{1}{y}) \cdot {(\frac{x}{y} + \frac{y}{x})}^{- 1}$
$= \frac{1}{x + y} \cdot \frac{x + y}{x y} \cdot \frac{x y}{x^{2} + y^{2}} = (x^{2} + y^{2})^{- 1}$

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