Question

If abc=1,prove that: $\frac{1}{1+a+b^{-1}} + \frac{1}{1+b+c^{-1}} + \frac{1}{1+c+a^{-1}} = 1$

Answer 1

$\frac{1}{1 + a + {b}^{ - 1} } + \frac{1}{1 + b + {c}^{ - 1} } + \frac{1}{1 + c + {a}^{ - 1} }$
$\frac{1}{1 + a + \frac{1}{b} } + \frac{1}{1 + b + \frac{1}{c} } + \frac{1}{1 + c + \frac{1}{a} }$
$\frac{b}{b + ab + 1} + \frac{c}{c + bc + 1} + \frac{a}{a + ac + 1}$
$\frac{b}{b + ab + 1} + \frac{c}{ \frac{1}{ab} + b \times \frac{1}{ab} + 1 } + \frac{a}{a + a \times \frac{1}{ab} +1}$
$\frac{b}{b + ab + 1} + \frac{ \frac{1}{ab} \times ab}{ \frac{1}{ab} \times ab + b \times \frac{1}{ab} \times ab + ab } + \frac{ab}{ab + 1 + b}$
$\frac{b}{b + ab + 1} + \frac{1}{b + ab + 1} + \frac{ab}{b + ab + 1}$
$\frac{b + ab + 1}{b + ab + 1}$
$= > 1 = 1$
hence verified