Example 14 Real numbers $a, b$ satisfy $ab=1$, let $M = \frac{1}{1+a} + \frac{1}{1+b}$, $N = \frac{a}{1+a} + \frac{b}{1+b}$. Then the relationship between $M$ and $N$ is $\qquad$
(A) $M>N$
(B) $M=N$
(C) $M<N$
(D) Uncertain
(1996, National Junior High School Mathematics Competition)
Solution: Replacing 1 in $N$ with $a b$ yields $N=\frac{a}{a b+a} +\frac{b}{a b+b}=\frac{1}{1+a}+\frac{1}{1+b}=M$. Therefore, the correct choice is (B).