\(\left(\frac{20052005}{20062006}-\frac{20042004}{20052005}\right)\cdot2005\cdot2006\)
=\(\left(\frac{2005\cdot10001}{2006\cdot10001}-\frac{2004\cdot10001}{2005\cdot10001}\right)\cdot2005\cdot2006\)
=\(\frac{2005\cdot2005\cdot2006}{2006}-\frac{2004\cdot2005\cdot2006}{2005}\)
= \(2005\cdot2005-2004\cdot2006\)
=\(2005^2-\left(2005-1\right)\cdot\left(2005+1\right)\)
=\(2005^2-2005^2+1\)
=1