Simplify. \( \frac{\frac{x}{y}-\frac{y}{x}}{\frac{1}{y}+\frac{1}{x}} \) \( \frac{\frac{x}{y}-\frac{y}{x}}{\frac{1}{y}+\frac{1}{x}}=\square \)

To simplify the expression \( \frac{\frac{x}{y}-\frac{y}{x}}{\frac{1}{y}+\frac{1}{x}} \), let's first focus on the numerator and denominator separately. The numerator \( \frac{x}{y} - \frac{y}{x} \) can be simplified by finding a common denominator, which is \( xy \): \[ \frac{x^2 - y^2}{xy} \] Now, we can factor \( x^2 - y^2 \) as \( (x-y)(x+y) \): \[ \frac{(x-y)(x+y)}{xy} \] Now, let's simplify the denominator \( \frac{1}{y} + \frac{1}{x} \): \[ \frac{x+y}{xy} \] Putting it all together, we have: \[ \frac{\frac{(x-y)(x+y)}{xy}}{\frac{x+y}{xy}} \] When we divide by a fraction, it’s the same as multiplying by its reciprocal: \[ \frac{(x-y)(x+y)}{xy} \cdot \frac{xy}{x+y} \] Here, the \( xy \) cancels out: \[ \frac{(x-y)(x+y)}{x+y} \] As long as \( x+y \neq 0 \), we can simplify further to: \[ x - y \] Finally, the simplified expression is: \[ \frac{\frac{x}{y}-\frac{y}{x}}{\frac{1}{y}+\frac{1}{x}} = x - y \]