Simplify $\frac{1}{1+\sqrt{2}}\cdot\frac{1}{1-\sqrt{2}}$.
Multiplying the numerators simply yields $1$. Multiplying the denominators gives $1+\sqrt{2} - \sqrt{2} -2 = 1 - 2 = -1$. So, the answer is $\frac{1}{-1} = \boxed{-1}$.