Prove that: a+b+c a^ -1 b^ -1 +b^ -1 c^ -1 +c^ -1 a^ -1 = abc - Vidyadip

Question

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

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Answer

$\ce{LHS}$
$=\frac{a+b+c}{a^{-1} b^{-1}+b^{-1} c^{-1}+c^{-1} a^{-1}}$
$=\frac{a+b+c}{\frac{1}{a} \times \frac{1}{b}+\frac{1}{b} \times \frac{1}{c}+\frac{1}{c} \times \frac{1}{a}}$
$=\frac{a+b+c}{\frac{1}{ab}+\frac{1}{b c}+\frac{1}{ca}}$
$=\frac{\frac{a+b+c}{c+a+b}}{a b c}$
$=\frac{a b c(a+b+c)}{a+b+c}$
$=a b c$
$=\text { RHS }$
$\ce{LHS=RHS}$
Hence Proved

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