\(\left(1-\dfrac{1}{2^2}\right)\left(1-\dfrac{1}{3^2}\right)\left(1-\dfrac{1}{4^2}\right)\left(1-\dfrac{1}{5^2}\right)...\left(1-\dfrac{1}{99^2}\right)\)
= \(\left(\dfrac{4}{4}-\dfrac{1}{4}\right)\left(\dfrac{9}{9}-\dfrac{1}{9}\right)\left(\dfrac{16}{16}-\dfrac{1}{16}\right)\left(\dfrac{25}{25}-\dfrac{1}{25}\right)...\left(\dfrac{9801}{9801}-\dfrac{1}{9801}\right)\)
= \(\dfrac{3}{4}.\dfrac{8}{9}.\dfrac{15}{16}.\dfrac{24}{25}.....\dfrac{9800}{9801}\)
=\(\dfrac{1.3}{2.2}.\dfrac{2.4}{3.3}.\dfrac{3.5}{4.4}.\dfrac{4.6}{5.5}.....\dfrac{98.100}{99.99}\)
=\(\dfrac{100}{2.99}=\dfrac{100}{198}\)
\(\left(1-\dfrac{1}{2^2}\right).\left(1-\dfrac{1}{3^2}\right).\left(1-\dfrac{1}{4^2}\right).\left(1-\dfrac{1}{5^2}\right).....\left(1-\dfrac{1}{99^2}\right)\)
\(=\dfrac{3}{2.2}.\dfrac{8}{3.3}.\dfrac{15}{4.4}.\dfrac{24}{5.5}.....\dfrac{9800}{99.99}\)
\(=\dfrac{1.3}{2.2}.\dfrac{2.4}{3.3}.\dfrac{3.5}{4.4}.\dfrac{4.6}{5.5}.....\dfrac{98.100}{99.99}\)
\(=\dfrac{\left(1.2.3.4.....98\right)}{\left(2.3.4.5.....99\right)}.\dfrac{\left(3.4.5.6.....100\right)}{\left(2.3.4.5.....99\right)}\)
\(=\dfrac{1}{99}.\dfrac{100}{2}\)
\(=\dfrac{50}{99}\)
