Answer:
Step-by-step explanation:
[tex]a^{-1}=\dfrac{1}{a} \ ; \ b^{-1}=\dfrac{1}{b} \ ; \ c^{-1}=\dfrac{1}{c}[/tex]
[tex]\dfrac{1}{1+a+b^{-1}}+\dfrac{1}{1+b+c^{-1}}+\dfrac{1}{1+c+a^{-1}}=\dfrac{1}{1+a+ \dfrac{1}{b}}+\dfrac{1}{1+b+\dfrac{1}{c}}+\dfrac{1}{1+c+\dfrac{1}{a}}[/tex]
[tex]= \dfrac{1}{\dfrac{1*b}{1*b}+\dfrac{a*b}{1*b}+\dfrac{1}{b}}+\dfrac{1}{\dfrac{1*c}{1*c}+\dfrac{b*c}{1*c}+\dfrac{1}{c}}+\dfrac{1}{\dfrac{1*a}{1*a}+\dfrac{c*a}{1*a}+\dfrac{1}{a}}\\\\\\=\dfrac{1}{\dfrac{b+ab+1}{b}}+\dfrac{1}{\dfrac{c+bc+1}{c}}+\dfrac{1}{\dfrac{a+ac+1}{a}}\\\\\\=\dfrac{b}{b+ab+1}+\dfrac{c}{c+bc+1}+\dfrac{a}{a+ac+1}[/tex]
abc = 1
c = 1/ab
[tex]=\dfrac{b}{b+ab+1}+\dfrac{\dfrac{1}{ab}}{\dfrac{1}{ab}+b*\dfrac{1}{ab}+1}+\dfrac{a}{a+a*\dfrac{1}{ab}+1}\\\\\\[/tex]
[tex]= \dfrac{b}{b+ab+1}+\dfrac{\dfrac{1}{ab}}{\dfrac{1+b+ab}{ab}}+\dfrac{a}{\dfrac{ab+1+b}{b}}\\\\\\=\dfrac{b}{b+ab+1}+\dfrac{\dfrac{1}{ab}*ab}{1+b+ab}+\dfrac{ab}{ab+1+b}[/tex]
[tex]=\dfrac{b}{b+ab+1}+\dfrac{1}{1+b+ab}+\dfrac{ab}{1+b+ab}\\\\\\=\dfrac{b+1+ab}{b+1+ab}\\\\\\=1[/tex]
