\begin{align} &\Big(1-\frac{1}{2^2}\Big)\Big(1-\frac{1}{3^2}\Big)\cdots\Big(1-\frac{1}{9^2}\Big)\Big(1-\frac{1}{10^2}\Big)\\ &=\Big(1-\frac{1}{2}\Big)\Big(1+\frac{1}{2}\Big)\Big(1-\frac{1}{3}\Big)\Big(1+\frac{1}{3}\Big)\cdots\Big(1-\frac{1}{9}\Big)\Big(1+\frac{1}{9}\Big)\Big(1-\frac{1}{10}\Big)\Big(1+\frac{1}{10}\Big)\\ &=\frac{1}{2}\cdot\frac{3}{2}\cdot\frac{2}{3}\cdot\frac{4}{3}\cdots\frac{8}{9}\cdot\frac{10}{9}\cdot\frac{9}{10}\cdot\frac{11}{10}\\ &=\boxed{\frac{11}{20}} \end{align}