Solve the following expression:
$$\frac{{\left(\frac{9}{8}+\frac{8}{9}\right)}^2-{\left(\frac{9}{8}-\frac{8}{9}\right)}^2}{{\left(\frac{8}{9}+\frac{9}{8}\right)}^2-{\left(\frac{8}{9}-\frac{9}{8}\right)}^2}$$

Let's use the standard algebraic identity:
$$(a+b{)}^2-(a-b{)}^2=4ab$$
In the numerator, let $a=\frac{9}{8}$ and $b=\frac{8}{9}$:
$$\text{Numerator}=4\times \left(\frac{9}{8}\right)\times \left(\frac{8}{9}\right)=4\times 1=4$$
In the denominator, let $a=\frac{8}{9}$ and $b=\frac{9}{8}$:
$$\text{Denominator}=4\times \left(\frac{8}{9}\right)\times \left(\frac{9}{8}\right)=4\times 1=4$$
Thus, the entire expression simplifies to:
$$\frac{4}{4}=1$$