Question
Prove that:
$\frac{\text{abc}}{\text{a}^{-1}\text{b}^{-1}+\text{b}^{-1}\text{c}^{-1}+\text{c}^{-1}\text{a}^{-1}}=\text{abc}$
$\frac{\text{abc}}{\text{a}^{-1}\text{b}^{-1}+\text{b}^{-1}\text{c}^{-1}+\text{c}^{-1}\text{a}^{-1}}=\text{abc}$
✓
Answer
$\frac{\text{abc}}{\text{a}^{-1}\text{b}^{-1}+\text{b}^{-1}\text{c}^{-1}+\text{c}^{-1}\text{a}^{-1}}=\text{abc}$
Left hand side (LHS) = Right hand side (RHS) Considering LHS,
$=\frac{\text{a}+\text{b}+\text{c}}{\frac{1}{\text{ab}}+\frac{1}{\text{bc}}+\frac{1}{\text{ca}}}$
$=\frac{\frac{\text{a}+\text{b}+\text{c}}{\text{a}+\text{b}+\text{c}}}{\text{abc}}$
$=\text{abc}$
Therefore, LHS = RHS Hence proved
Left hand side (LHS) = Right hand side (RHS) Considering LHS,
$=\frac{\text{a}+\text{b}+\text{c}}{\frac{1}{\text{ab}}+\frac{1}{\text{bc}}+\frac{1}{\text{ca}}}$
$=\frac{\frac{\text{a}+\text{b}+\text{c}}{\text{a}+\text{b}+\text{c}}}{\text{abc}}$
$=\text{abc}$
Therefore, LHS = RHS Hence proved
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