Simplify (1/x-1/y)/(1/(x^2)+1/(y^2))

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

Multiply the numerator and denominator of the complex fraction by $y^{2} x^{2}$ .

Multiply $\frac{\frac{1}{x} - \frac{1}{y}}{\frac{1}{x^{2}} + \frac{1}{y^{2}}}$ by $\frac{y^{2} x^{2}}{y^{2} x^{2}}$ .

$\frac{y^{2} x^{2}}{y^{2} x^{2}} \cdot \frac{\frac{1}{x} - \frac{1}{y}}{\frac{1}{x^{2}} + \frac{1}{y^{2}}}$

Combine.

$\frac{y^{2} x^{2} (\frac{1}{x} - \frac{1}{y})}{y^{2} x^{2} (\frac{1}{x^{2}} + \frac{1}{y^{2}})}$

Apply the distributive property.

$\frac{y^{2} x^{2} \frac{1}{x} + y^{2} x^{2} (- \frac{1}{y})}{y^{2} x^{2} \frac{1}{x^{2}} + y^{2} x^{2} \frac{1}{y^{2}}}$

Simplify by cancelling.

Cancel the common factor of $x$ .

Factor $x$ out of $y^{2} x^{2}$ .

$\frac{x (y^{2} x) \frac{1}{x} + y^{2} x^{2} (- \frac{1}{y})}{y^{2} x^{2} \frac{1}{x^{2}} + y^{2} x^{2} \frac{1}{y^{2}}}$

Cancel the common factor.

$\frac{x (y^{2} x) \frac{1}{x} + y^{2} x^{2} (- \frac{1}{y})}{y^{2} x^{2} \frac{1}{x^{2}} + y^{2} x^{2} \frac{1}{y^{2}}}$

Rewrite the expression.

$\frac{y^{2} x + y^{2} x^{2} (- \frac{1}{y})}{y^{2} x^{2} \frac{1}{x^{2}} + y^{2} x^{2} \frac{1}{y^{2}}}$

Cancel the common factor of $y$ .

Move the leading negative in $- \frac{1}{y}$ into the numerator.

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

Factor $y$ out of $y^{2} x^{2}$ .

$\frac{y^{2} x + y (y x^{2}) \frac{- 1}{y}}{y^{2} x^{2} \frac{1}{x^{2}} + y^{2} x^{2} \frac{1}{y^{2}}}$

Cancel the common factor.

$\frac{y^{2} x + y (y x^{2}) \frac{- 1}{y}}{y^{2} x^{2} \frac{1}{x^{2}} + y^{2} x^{2} \frac{1}{y^{2}}}$

Rewrite the expression.

$\frac{y^{2} x + y x^{2} \cdot - 1}{y^{2} x^{2} \frac{1}{x^{2}} + y^{2} x^{2} \frac{1}{y^{2}}}$

Cancel the common factor of $x^{2}$ .

Factor $x^{2}$ out of $y^{2} x^{2}$ .

$\frac{y^{2} x + y x^{2} \cdot - 1}{x^{2} y^{2} \frac{1}{x^{2}} + y^{2} x^{2} \frac{1}{y^{2}}}$

Cancel the common factor.

$\frac{y^{2} x + y x^{2} \cdot - 1}{x^{2} y^{2} \frac{1}{x^{2}} + y^{2} x^{2} \frac{1}{y^{2}}}$

Rewrite the expression.

$\frac{y^{2} x + y x^{2} \cdot - 1}{y^{2} + y^{2} x^{2} \frac{1}{y^{2}}}$

Cancel the common factor of $y^{2}$ .

Factor $y^{2}$ out of $y^{2} x^{2}$ .

$\frac{y^{2} x + y x^{2} \cdot - 1}{y^{2} + y^{2} (x^{2}) \frac{1}{y^{2}}}$

Cancel the common factor.

$\frac{y^{2} x + y x^{2} \cdot - 1}{y^{2} + y^{2} x^{2} \frac{1}{y^{2}}}$

Rewrite the expression.

$\frac{y^{2} x + y x^{2} \cdot - 1}{y^{2} + x^{2}}$

Simplify the numerator.

Factor $y x$ out of $y^{2} x + y x^{2} \cdot - 1$ .

Factor $y x$ out of $y^{2} x$ .

$\frac{y x (y) + y x^{2} \cdot - 1}{y^{2} + x^{2}}$

Factor $y x$ out of $y x^{2} \cdot - 1$ .

$\frac{y x (y) + y x (x \cdot - 1)}{y^{2} + x^{2}}$

Factor $y x$ out of $y x (y) + y x (x \cdot - 1)$ .

$\frac{y x (y + x \cdot - 1)}{y^{2} + x^{2}}$

Move $- 1$ to the left of $x$ .

$\frac{y x (y - 1 \cdot x)}{y^{2} + x^{2}}$

Rewrite $- 1 x$ as $- x$ .

$\frac{y x (y - x)}{y^{2} + x^{2}}$

Simplify (1/x-1/y)/(1/(x^2)+1/(y^2))

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