Question

Show that
$\frac{ {x}^{ - 1} + {y}^{ - 1} } {x ^{ - 1} } + \frac{ {x}^{ - 1} - y ^{ -1 } }{x ^{ - 1} } = \frac{ {x}^{2} + {y}^{2} }{xy}$
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Answer 1

$\frac{ {x}^{ - 1} + {y}^{ - 1} }{ {x}^{ - 1} } + \frac{ {x}^{ - 1} - {y}^{ - 1} }{ {y}^{ - 1} } \\ = \frac{ \frac{1}{x} + \frac{1}{y} }{ \frac{1}{x} } + \frac{ \frac{1}{x} - \frac{1}{y} }{ \frac{1}{y} } \\ = \frac{x + y}{xy} \times x + \frac{y - x}{xy} \times y \\ = \frac{x + y}{y} + \frac{y - x}{x} \\ = \frac{ {x}^{2} + xy + {y}^{2} - xy }{xy} \\ = \frac{ {x}^{2} + {y}^{2} }{xy}$