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

Given:

\( a b c=1 \)

To do:

We have to show that \( \frac{1}{1+a+b^{-1}}+\frac{1}{1+b+c^{-1}}+\frac{1}{1+c+a^{-1}}=1 \).

Solution:

We know that,

$(a^{m})^{n}=a^{m n}$

$a^{m} \times a^{n}=a^{m+n}$

$a^{m} \div a^{n}=a^{m-n}$

$a^{0}=1$  

$abc=1$

$\Rightarrow c=\frac{1}{ab}$.....(i)

$ab=\frac{1}{c}$........(ii)

LHS $=\frac{1}{1+a+b^{-1}}+\frac{1}{1+b+c^{-1}}+\frac{1}{1+c+a^{-1}}$

$=\frac{1}{1+a+\frac{1}{b}}+\frac{1}{1+b+\frac{1}{c}}+\frac{1}{1+c+\frac{1}{a}}$

$=\frac{1}{\frac{b+ab+1}{b}}+\frac{1}{1+b+ab}+\frac{1}{1+\frac{1}{ab}+\frac{1}{a}}$        [From (i) and (ii)]

$=\frac{b}{b+a b+1}+\frac{1}{1+b+a b}+\frac{a b}{a b+1+b}$

$=\frac{b+1+a b}{b+1+a b}$

$=1$

Hence proved.      

Tutorialspoint

Updated on: 10-Oct-2022

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