Question

Prove that:

(i) $\frac{a+b+c}{a^{-1}{b}^{-1}+b^{-1}{c}^{-1}+c^{-1}{a}^{-1}}=abc$

(ii) ${\left(a^{-1}+b^{-1}\right)}^{-1}=\frac{ab}{a+b}$

Answer 1

(i) Consider the left hand side:
$$\begin{gathered} \frac{a+b+c}{a^{-1}{b}^{-1}+b^{-1}{c}^{-1}+c^{-1}{a}^{-1}} \\ =\frac{a+b+c}{{\displaystyle \frac{1}{ab}}+{\displaystyle \frac{1}{bc}}+{\displaystyle \frac{1}{ca}}} \\ =\frac{a+b+c}{{\displaystyle \frac{c+a+b}{abc}}} \\ =\left(a+b+c\right)\times \left(\frac{abc}{a+b+c}\right) \\ =abc \end{gathered}$$
Therefore left hand side is equal to the right hand side. Hence proved.

(ii)
Consider the left hand side:
$$\begin{gathered} {\left(a^{-1}+b^{-1}\right)}^{-1} \\ =\frac{1}{a^{-1}+b^{-1}} \\ =\frac{1}{{\displaystyle \frac{1}{a}}+{\displaystyle \frac{1}{b}}} \\ =\frac{1}{{\displaystyle \frac{b+a}{ab}}} \\ =\frac{ab}{a+b} \end{gathered}$$
Therefore left hand side is equal to the right hand side. Hence proved.