Question 1023790
Use a substitution,
{{{X=a-b}}}
{{{Y=a+b}}}
So then,
{{{(a-b)/(a+b)-(a+b)/(a-b)=X/Y-Y/X}}}
You want a common denominator {{{XY}}}
{{{(a-b)/(a+b)-(a+b)/(a-b)=(X/Y)(X/X)-(Y/X)(Y/Y)}}}

{{{(a-b)/(a+b)-(a+b)/(a-b)=X^2/(XY)-Y^2/(XY)}}}
{{{(a-b)/(a+b)-(a+b)/(a-b)=(X^2-Y^2)/(XY)}}}
So,
{{{X^2=(a-b)^2=a^2-2ab+b^2}}}
{{{Y^2=(a+b)^2=a^2+2ab+b^2}}}
{{{X^2-Y^2=a^2-2ab+b^2-a^2-2ab-b^2}}}
{{{X^2-Y^2=-4ab}}}
and
{{{XY=(a-b)(a+b)=a^2-b^2}}}
Substituting,
{{{(a-b)/(a+b)-(a+b)/(a-b)=(-4ab)/(a^2-b^2)}}}