Prove that:

(i) Consider the left hand side:

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

$=\frac{a+b+c}{\frac{1}{a b}+\frac{1}{b c}+\frac{1}{c a}}$

$=\frac{a+b+c}{\frac{c+a+b}{a b c}}$

$=(a+b+c) \times\left(\frac{a b c}{a+b+c}\right)$

$=a b c$

Therefore left hand side is equal to the right hand side. Hence proved.

(ii)
Consider the left hand side:

$\left(a^{-1}+b^{-1}\right)^{-1}$

$=\frac{1}{a^{-1}+b^{-1}}$

$=\frac{1}{\frac{1}{a}+\frac{1}{b}}$

$=\frac{1}{\frac{b+a}{a b}}$

$=\frac{a b}{a+b}$

Therefore left hand side is equal to the right hand side. Hence proved.