Simplify each complex fraction. See Examples 5 and 6. (x/y + y/x) / (x/y − y/x)

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Rewrite the complex fraction clearly as a division of two fractions: the numerator is \(\frac{x}{y} + \frac{y}{x}\) and the denominator is \(\frac{x}{y} - \frac{y}{x}\).

Find a common denominator for the fractions in both the numerator and denominator. For each, the common denominator is \(xy\), so rewrite each fraction accordingly: \(\frac{x}{y} = \frac{x^2}{xy}\) and \(\frac{y}{x} = \frac{y^2}{xy}\).

Combine the fractions in the numerator and denominator separately by adding or subtracting the numerators over the common denominator \(xy\): numerator becomes \(\frac{x^2 + y^2}{xy}\) and denominator becomes \(\frac{x^2 - y^2}{xy}\).

Rewrite the complex fraction as a division of two fractions: \(\frac{\frac{x^2 + y^2}{xy}}{\frac{x^2 - y^2}{xy}}\). Then, apply the rule for dividing fractions by multiplying the numerator by the reciprocal of the denominator.

Simplify the resulting expression by canceling the common denominator \(xy\) and express the final simplified form as \(\frac{x^2 + y^2}{x^2 - y^2}\).

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