(1-1/2^2)*(1-1/3^2)*(1-1/4^2)*.*(1-1/2002^2)*(1-1/2003^2)

(1-1/2^2)*(1-1/3^2)*(1-1/4^2)*.*(1-1/2002^2)*(1-1/2003^2)
(1-1/2^2)*(1-1/3^2)*(1-1/4^2)*.*(1-1/2002^2)*(1-1/2003^2)

(1-1/2^2)*(1-1/3^2)*(1-1/4^2)*.*(1-1/2002^2)*(1-1/2003^2)
(1-1/2^2)=1/2*3/2
(1-1/3^2)=2/3*4/3
.
(1-1/2^2)*(1-1/3^2)*(1-1/4^2)*.*(1-1/2002^2)*(1-1/2003^2)
=1/2*3/2*2/3*4/3*3/4*5/4*.*2002/2003*2004/2003
=1/2*2004/2003
=1002/2003

(1-1/2^2)*(1-1/3^2)*(1-1/4^2)*......*(1-1/2002^2)*(1-1/2003^2)
=(1-1/2)*(1+1/2)*(1-1/3)*(1+1/3)*(1-1/4)(1+1/4)*……*(1-1/2002)(1+1/2002)*(1-1/2003)*(1+1/2003)
=(1/2)*(3/2)*(2/3)*(4/3)*(3/4)*(5/4)*……*(2001/2002)*(2003/2002)*(2002/2003)*(
=(1/2)*(2004/2003)
=1002/2003