The problem asks us to simplify the given rational expression:

$\frac{\frac{x}{y} - \frac{y}{x}}{\frac{x}{y} + \frac{y}{x}}$

Steps to Simplify:

1. Simplify the numerator ($\frac{x}{y} - \frac{y}{x}$):

   • Find a common denominator for $\frac{x}{y}$ and $\frac{y}{x}$, which is $xy$: $\frac{x}{y} - \frac{y}{x} = \frac{x^2 - y^2}{xy}$

2. Simplify the denominator ($\frac{x}{y} + \frac{y}{x}$):

   • Again, find a common denominator for $\frac{x}{y}$ and $\frac{y}{x}$, which is $xy$: $\frac{x}{y} + \frac{y}{x} = \frac{x^2 + y^2}{xy}$

3. Combine the numerator and denominator into a single fraction: $\frac{\frac{x^2 - y^2}{xy}}{\frac{x^2 + y^2}{xy}}$

4. Simplify the complex fraction:

   • Multiply by the reciprocal of the denominator: $\frac{x^2 - y^2}{xy} \div \frac{x^2 + y^2}{xy} = \frac{x^2 - y^2}{x^2 + y^2}$

Final Answer:

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

Would you like a detailed explanation for any step or further assistance with similar problems?

Related Questions:

1. How do you simplify expressions with variables and fractions?
2. What is the difference of squares, and how does it apply to $x^2 - y^2$?
3. Can the denominator $x^2 + y^2$ ever be factored further?
4. How does this method change if additional variables are introduced?
5. How do you verify the simplification step-by-step?

Tip: Always check for common denominators early when dealing with complex fractions to simplify your work!