Given,

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

Multiplying and dividing the first term by $\rm (x + y) $and the second term by $\rm (x - y)$, we get,

$\rm = \frac{x + y}{x - y} \cdot \frac{x + y}{x + y} - \frac{x - y}{x + y} \cdot \frac{x - y}{x - y}$

Taking LCM and simplifying the expression, we get,

$\rm = \frac{ (x + y)(x + y) - (x - y)(x - y)}{ (x - y)(x + y)}$

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

Using the expansion formula for each of the terms in the above expression, we get,

$\rm = \frac{ (x^{2} + 2xy + y^{2}) - ( x^{2} - 2xy + y^{2} ) }{ x^{2} - y^{2}}$

$\rm = \frac{ x^{2} + 2xy + y^{2} - x^{2} + 2xy - y^{2} }{x^{2} - y^{2}}$

$\rm = \frac{ 4xy}{x^{2} - y^{2}}$

$\rm \therefore \frac{x + y}{x - y} - \frac{x - y}{x + y} = \frac{ 4xy}{x^{2} - y^{2}}$