\(\left(1-\dfrac{1}{2^2}\right)\left(1-\dfrac{1}{3^2}\right)\left(1-\dfrac{1}{4^2}\right)...\left(1-\dfrac{1}{1999^2}\right)\left(1-\dfrac{1}{2000^2}\right)=\dfrac{1\times3}{2\times2}\times\dfrac{2\times4}{3\times3}\times\dfrac{3\times5}{4\times4}\times...\times\dfrac{1999\times2001}{2000\times2000}\)

\(=\dfrac{\left(1\times2\times3\times...\times1999\right)\left(3\times4\times5\times...\times2001\right)}{\left(2\times3\times4\times...\times2000\right)\left(2\times3\times4\times...\times2000\right)}\)

\(=\dfrac{1\times2001}{2000\times2}=\dfrac{2001}{4000}\)