Question

If $x=\frac{a+b}{a-b}$ and $y=\frac{a-b}{a+b}$, find $x-y$.

Answer 1

We know the identities $a^2-b^2=(a+b)(a-b)$ and $(a+b{)}^2=a^2+b^2+2ab$

It is given that $x=\frac{a+b}{a-b}$ and $y=\frac{a-b}{a+b}$ and calculate $x-y$ using above identities as shown below:

$x-y=\frac{a+b}{a-b}-\frac{a-b}{a+b}=\frac{(a+b)(a+b)-(a-b)(a-b)}{(a-b)(a+b)}=\frac{(a+b{)}^2-(a-b{)}^2}{a^2-b^2}=\frac{(a^2+b^2+2ab)-(a^2+b^2-2ab)}{a^2-b^2}$

$=\frac{a^2+b^2+2ab-a^2-b^2+2ab}{a^2-b^2}=\frac{4ab}{a^2-b^2}$

Hence, $x-y=\frac{4ab}{a^2-b^2}$