1. **State the problem:** Given the equation $x^2 + y^2 = 1$, show that $$\frac{\frac{x}{y}}{1 + \frac{y^2}{x^2}} = xy.$$\n\n2. **Rewrite the expression:** The left side is a complex fraction. Start by simplifying the denominator:\n$$1 + \frac{y^2}{x^2} = \frac{x^2}{x^2} + \frac{y^2}{x^2} = \frac{x^2 + y^2}{x^2}.$$\n\n3. **Substitute the denominator back:**\n$$\frac{\frac{x}{y}}{1 + \frac{y^2}{x^2}} = \frac{\frac{x}{y}}{\frac{x^2 + y^2}{x^2}}.$$\n\n4. **Divide by a fraction by multiplying by its reciprocal:**\n$$= \frac{x}{y} \times \frac{x^2}{x^2 + y^2}.$$\n\n5. **Use the given equation $x^2 + y^2 = 1$ to simplify:**\n$$= \frac{x}{y} \times \frac{x^2}{1} = \frac{x}{y} \times x^2 = \frac{x^3}{y}.$$\n\n6. **Check if this equals $xy$:** The expression $\frac{x^3}{y}$ is not equal to $xy$ in general.\n\n7. **Re-examine the problem:** Possibly the problem means to show the expression simplifies to $xy$. Let's simplify the original expression differently:\n\nRewrite the original expression as $$\frac{\frac{x}{y}}{1 + \frac{y^2}{x^2}} = \frac{\frac{x}{y}}{\frac{x^2 + y^2}{x^2}} = \frac{x}{y} \times \frac{x^2}{x^2 + y^2} = \frac{x^3}{y(x^2 + y^2)}.$$\n\nSince $x^2 + y^2 = 1$, substitute:\n$$= \frac{x^3}{y \times 1} = \frac{x^3}{y}.$$\n\nThis is not equal to $xy$ unless $x^2 = y^2$, which is not generally true.\n\n8. **Check the original expression carefully:** The problem states $\frac{\frac{x}{y}}{1 + \frac{y^2}{x^2}} = xy$. Let's rewrite the denominator as a single fraction:\n$$1 + \frac{y^2}{x^2} = \frac{x^2 + y^2}{x^2} = \frac{1}{x^2}.$$\n\nThis is incorrect because $x^2 + y^2 = 1$, so denominator is $\frac{1}{x^2}$.\n\n9. **Therefore:**\n$$\frac{\frac{x}{y}}{1 + \frac{y^2}{x^2}} = \frac{\frac{x}{y}}{\frac{1}{x^2}} = \frac{x}{y} \times x^2 = \frac{x^3}{y}.$$\n\n10. **Try to rewrite $\frac{x^3}{y}$ as $xy$:** Multiply numerator and denominator by $y$:\n$$\frac{x^3}{y} = x^3 \times \frac{1}{y} = x \times x^2 \times \frac{1}{y}.$$\n\nNo direct simplification to $xy$.\n\n11. **Conclusion:** The original expression simplifies to $\frac{x^3}{y}$, not $xy$. Possibly the problem has a typo or different expression.\n\n**Final answer:** $$\frac{\frac{x}{y}}{1 + \frac{y^2}{x^2}} = \frac{x^3}{y}.$$