<s>[INST] Basic algebra problem: $ \frac{\frac{1}{x}+\frac{1}{y}}{\frac{1}{x^2}-\frac{1}{y^2}} $ Basic algebra problem I can't seem to figure out: $$ \frac{\frac{1}{x}+\frac{1}{y}}{\frac{1}{x^2}-\frac{1}{y^2}} $$ $x,y \in \mathbb{R}, x^2 \neq y^2, xy\neq0$.
Now I know the result is: $\frac{xy}{y-x}$, but I am not sure how to get it, I get into a mess like this: $=x+\frac{x^2}{y}-\frac{y^2}{x}-y=\frac{x(xy)+x^3-y^3-y(xy)}{xy}=?$ which doesn't seem to help me much. Halp please.
 [/INST] $$a=\frac1x+\frac1y=\frac{x+y}{xy}$$
$$b=\frac1{x^2}-\frac1{y^2}=\frac{y^2-x^2}{(xy)^2}$$
$$\frac{a}b=\frac{(x+y)xy}{x^2-y^2}=\frac{(x+y)xy}{(y-x)(y+x)}$$
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