Answer
$x-\dfrac{y}{\dfrac{x}{y}+\dfrac{y}{x}}=\dfrac{x^{3}}{x^{2}+y^{2}}$
Work Step by Step
$x-\dfrac{y}{\dfrac{x}{y}+\dfrac{y}{x}}$
Evaluate the sum in the denominator:
$x-\dfrac{y}{\dfrac{x}{y}+\dfrac{y}{x}}=x-\dfrac{y}{\Big(\dfrac{x^{2}+y^{2}}{xy}\Big)}=...$
Evaluate the division:
$...=x-\dfrac{(xy)y}{x^{2}+y^{2}}=x-\dfrac{xy^{2}}{x^{2}+y^{2}}=...$
Finally, evaluate the substraction and simplify:
$...=\dfrac{x(x^{2}+y^{2})-xy^{2}}{x^{2}+y^{2}}=\dfrac{x^{3}+xy^{2}-xy^{2}}{x^{2}+y^{2}}=\dfrac{x^{3}}{x^{2}+y^{2}}$
