(1-1/2^2)(1-1/3^2)(1-1/4^2)...(1-1/2008^2)
=(1+1/2)(1-1/2)(1+1/3)(1-1/3)(1+1/4)(1-1/4)...(1+1/2008)(1-1/2008)
=[(1+1/2)(1+1/3)(1+1/4)...(1+1/2008)][(1-1/2)(1-1/3)(1-1/4)...(1-1/2008)]
=[3/2*4/3*5/4*...*2009/2008][1/2*2/3*3/4*...*2007/2008]
=2009/2*1/2008
=2009/4016.