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

To simplify the given expression, let's first simplify the numerator and the denominator separately. For the numerator \( \frac{x}{y} - \frac{y}{x} \): 1. Find a common denominator, which is \( xy \). 2. Rewrite it: \[ \frac{x^2 - y^2}{xy} \] For the denominator \( \frac{1}{y} + \frac{1}{x} \): 1. Again, use the common denominator \( xy \): \[ \frac{x + y}{xy} \] Now, substitute these simplified forms back into the original expression: \[ \frac{\frac{x^2 - y^2}{xy}}{\frac{x + y}{xy}} \] The \( xy \) in the numerator and denominator cancels out: \[ \frac{x^2 - y^2}{x + y} \] Now, recognize that \( x^2 - y^2 \) can be factored: \[ \frac{(x - y)(x + y)}{x + y} \] Assuming \( x + y \neq 0 \) (to avoid division by zero), these can be simplified: \[ x - y \] So, the simplified expression is: \( x - y \)