Find the value of x $$\left(1-\frac{1}{2^2}\right)\left(1-\frac{1}{3^2}\right)\left(1-\frac{1}{4^2}\right)\ldots\left(1-\frac{1}{2011^2}\right)=\frac{x}{2\cdot 2012}$$
Help me solve this, obviously there is a shortcut to solve it, but I have tried a lot and I've come up with something confusing!

If you use the hint (given by @Harald Hanche-Olsen) and write out some terms you will see a pattern emerging:$$\require{cancel}\frac{1.\cancel{3}}{2.\cancel{2}}\frac{\cancel{2}.\cancel{4}}{\cancel{3}.\cancel{3}}\frac{\cancel{3}.5}{\cancel{4}.4}...$$Can you see how all the intermediate terms will cancel out?
