The Simplified value of $$  \left(1 - \frac{2xy}{x^2 + y^2}\right) \div \left(\frac{x^3 - y^3}{x - y} - 3xy\right)$$ is

$$  \left(1 - \frac{2xy}{x^2 + y^2}\right) \div \left(\frac{x^3 - y^3}{x - y} - 3xy\right)$$

= $$\left(\frac{x^2 + y^2-2xy}{x^2 + y^2}\right) \div \left(\frac{x^3 - y^3 -3xy(x - y)}{x - y} \right)$$ 

($$\because x^2 + y^2-2xy = (x-y)^2$$  and $$x^3 - y^3 -3xy(x - y) = (x-y)^3$$)

=$$\left(\frac{(x-y)^2}{x^2 + y^2}\right) \div \left(\frac{(x- y)^3}{x - y} \right)$$

= $$\left(\frac{(x-y)^2}{x^2 + y^2}\right) \times \left(\frac{{x - y}}{(x- y)^3}\right)$$ 

=$$\frac{1}{x^2 + y^2}$$