Question
Prove that $: \frac{a+b+c}{a^{-1} b^{-1}+b^{-1} c^{-1}+c^{-1} a^{-1}}=a b c$
✓
Answer
$ \text { L.H.S. }=\frac{a+b+c}{a^{-1} b^{-1}+b^{-1} c^{-1}+c^{-1} a^{-1}}$
$ =\frac{a+b+c}{\frac{1}{a b}+\frac{1}{b c}+\frac{1}{c a}}$
$ =\frac{a+b+c}{\frac{c+a+b}{a b c}}$
$ =\frac{(a+b+c)(a b c)}{a+b+c}$
$ =a b c$
$ =\text { R.H.S. }$
$ =\frac{a+b+c}{\frac{1}{a b}+\frac{1}{b c}+\frac{1}{c a}}$
$ =\frac{a+b+c}{\frac{c+a+b}{a b c}}$
$ =\frac{(a+b+c)(a b c)}{a+b+c}$
$ =a b c$
$ =\text { R.H.S. }$
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