Answer
$\frac{x^{3}}{x^{2}+y^{2}}$
Work Step by Step
We multiply the right fraction by $\frac{xy}{xy}$. Then we form a common denominator and combine the fractions:
$\displaystyle x-\frac{y}{\frac{x}{y}+\frac{y}{x}}=x-\frac{y}{\frac{x}{y}+\frac{y}{x}}\cdot\frac{xy}{xy}=x-\frac{xy^{2}}{x^{2}+y^{2}}=\frac{x(x^{2}+y^{2})}{x^{2}+y^{2}}-\frac{xy^{2}}{x^{2}+y^{2}}=\frac{x^{3}+xy^{2}-xy^{2}}{x^{2}+y^{2}}=\frac{x^{3}}{x^{2}+y^{2}}$
