Question

if abc = 1 , prove that: 1/1+a+b^-1 + 1/1+b+c^-1 + 1/1+c+a^-1​

Answer 1

$\bigstar \underline{\underline{\bf Question:-}}$

• If abc = 1, show that 1/(1+a+b^-1) + 1/(1+b+c^-1) + 1/(1+c+a^-1) = 1

$\bigstar \underline{\underline{\bf Proof:-}}$

$\rightarrow \sf a^{-1}=\frac{1}{a} ;b^{-1} = \frac{1}{b} ; c^{-1}=\frac{1}{c}$

Given that abc = 1 .

=> c = 1/ab

$\\ :\implies \sf \frac{1}{1+a+b^{-1}} +\frac{1}{1+b+c^{-1}} +\frac{1}{1+c+a^{-1}}$

$:\implies \sf \frac{1}{1+a+\frac{1}{b}} +\frac{1}{1+b+\frac{1}{c} } +\frac{1}{1+c+\frac{1}{a} }$

$:\implies \sf \frac{b}{b+ab+1} +\frac{c}{c+bc+1} +\frac{a}{a+ac+1}$

As c=1/ab,

$\\ :\implies \sf \frac{b}{b+ab+1} +\frac{1/ab}{(1/ab)+b(1/ab)+1} +\frac{a}{a+a(1/ab)+1}$

$:\implies \sf \frac{b}{b+ab+1} +\frac{\frac{1}{ab} \times ab}{1+b+ab} +\frac{ab}{1+ab+b}$

$:\implies \sf \frac{b}{b+ab+1} +\frac{1}{1+b+ab} +\frac{ab}{1+ab+b}$

$:\implies \sf \frac{1+ab+b}{b+ab+1} =\pink{\underline{\bold{1 .}}}$

Hence proved !

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Answer 2

$\bigstar \underline{\underline{\bf Question:-}}$

If abc = 1, show that 1/(1+a+b^-1) + 1/(1+b+c^-1) + 1/(1+c+a^-1) = 1

$\bigstar \underline{\underline{\bf Proof:-}}$

$\rightarrow \sf a^{-1}=\frac{1}{a} ;b^{-1} = \frac{1}{b} ; c^{-1}=\frac{1}{c}$

Given that abc = 1 .

=> c = 1/ab

$\\ :\implies \sf \frac{1}{1+a+b^{-1}} +\frac{1}{1+b+c^{-1}} +\frac{1}{1+c+a^{-1}}$

$:\implies \sf \frac{1}{1+a+\frac{1}{b}} +\frac{1}{1+b+\frac{1}{c} } +\frac{1}{1+c+\frac{1}{a} }$

$:\implies \sf \frac{b}{b+ab+1} +\frac{c}{c+bc+1} +\frac{a}{a+ac+1}$

As c=1/ab,

$\\ :\implies \sf \frac{b}{b+ab+1} +\frac{1/ab}{(1/ab)+b(1/ab)+1} +\frac{a}{a+a(1/ab)+1}$

$:\implies \sf \frac{b}{b+ab+1} +\frac{\frac{1}{ab} \times ab}{1+b+ab} +\frac{ab}{1+ab+b}$

$:\implies \sf \frac{b}{b+ab+1} +\frac{1}{1+b+ab} +\frac{ab}{1+ab+b}$

$:\implies \sf \frac{1+ab+b}{b+ab+1} =\pink{\underline{\bold{1 .}}}$

Hence proved !

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