Prove that:

(i) To prove,

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

Left hand side (LHS) = Right hand side (RHS) Considering LHS,

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

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

= abc

Therefore, LHS = RHS Hence proved

(ii) To prove,

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

Left hand side (LHS) = Right hand side (RHS) Considering LHS,

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

$=\frac{1}{\left(\frac{1}{a}+\frac{1}{b}\right)}$

$=\frac{1}{\left(\frac{a+b}{a b}\right)}$

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

Therefore, LHS = RHS

Hence proved