$$\frac{x}{x+y}+\frac{y}{x-y}=?$$

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

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

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