Given

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

Using the factorization formula for $\rm (x^{2} - y^{2}) = (x + y)(x - y)$, we get,

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

$\rm = \frac{a}{x-y} - \frac{1}{ x - y}$

$\rm = \frac{a - 1}{x - y}$

$\rm \therefore \frac{a}{x - y} - \frac{x + y}{x^{2} - y^{2}}= \frac{a - 1}{x - y}$