$(1-\dfrac{1}{2^2}).(1-\dfrac{1}{3^2}).(1-\dfrac{1}{4^2})...(1-\dfrac{1}{9^2})$
$=(1-\dfrac{1}{4}).(1-\dfrac{1}{9}).(1-\dfrac{1}{16})...(1-\dfrac{1}{81})$
$=\dfrac{3}{4}.\dfrac{8}{9}.\dfrac{15}{16}...\dfrac{80}{81}$
$=\dfrac{1.3}{2.2}.\dfrac{2.4}{3.3}.\dfrac{3.5}{4.4}...\dfrac{8.10}{9.9}$
$=\dfrac{1.3.2.4.3.5...8.10}{2.2.3.3.4.4.9.9}$
$=\dfrac{5}{216}$