$\rightarrow\left(\frac{x+y}{\frac{1}{x} + \frac{1}{y}}\right)\left(\frac{xy}{xy}\right) \rightarrow \left(\frac{x+y}{\frac{xy}{x} + \frac{xy}{y}}\right)\left(\frac{xy}{1}\right) \rightarrow \left(\frac{x+y}{x+y}\right)\left(\frac{xy}{1}\right) \rightarrow xy$

$\rightarrow \left(\frac{x}{x+y} \left(\frac{x}{x}\right) - \frac{x-y}{x} \left(\frac{x+y}{x+y} \right) \right)$

$\rightarrow \frac{x^2}{x(x+y)}-\frac{(x-y)(x+y)}{x(x+y)}$

$\rightarrow\frac{x^2 - (x-y)(x+y)}{x(x+y)}$

$\rightarrow\frac{x^2-(x^2-y^2)}{x(x+y)}\rightarrow\frac{y^2}{x(x+y)}$