Chapter 6 - Section 6.3 - Simplifying Complex Fractions - Exercise Set - Page 362: 65

$\dfrac{x+2}{x+1}$

The given expression, $ \dfrac{1}{1+(1+x)^{-1}} ,$ simplifies to \begin{array}{l}\require{cancel} \dfrac{1}{1+\dfrac{1}{1+x}} \\\\= \dfrac{1}{1+\dfrac{1}{1+x}} \\\\= \dfrac{1}{\dfrac{(1+x)+1}{1+x}} \\\\= \dfrac{1}{\dfrac{x+2}{1+x}} \\\\= 1\div\dfrac{x+2}{1+x} \\\\= 1\cdot\dfrac{x+2}{1+x} \\\\= \dfrac{x+2}{1+x} \\\\= \dfrac{x+2}{x+1} .\end{array}