Taking out factors $a, b, c$ common from $\mathrm{R}_{1}, \mathrm{R}_{2}$ and $\mathrm{R}_{3},$ we get

$\text { L.H.S. }=a b c\left|\begin{array}{ccc}
\frac{1}{a}+1 & \frac{1}{a} & \frac{1}{a} \\
\frac{1}{b} & \frac{1}{b}+1 & \frac{1}{b} \\
\frac{1}{c} & \frac{1}{c} & \frac{1}{c}+1
\end{array}\right|$

Applying $\mathrm{R}_{1} \rightarrow \mathrm{R}_{1}+\mathrm{R}_{2}+\mathrm{R}_{3},$ we have

$\Delta=a b c\left|\begin{array}{ccc}
1+\frac{1}{a}+\frac{1}{b}+\frac{1}{c} & 1+\frac{1}{a}+\frac{1}{b}+\frac{1}{c} & 1+\frac{1}{a}+\frac{1}{b}+\frac{1}{c} \\
\frac{1}{b} & \frac{1}{b}+1 & \frac{1}{b} \\
\frac{1}{c} & \frac{1}{c} & \frac{1}{c}+1
\end{array}\right|$

$=a b c\left(1+\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\right)\left|\begin{array}{ccc}
1 & 1 & 1 \\
\frac{1}{b} & \frac{1}{b}+1 & \frac{1}{b} \\
\frac{1}{c} & \frac{1}{c} & \frac{1}{c}+1
\end{array}\right|$

Now applying $\mathrm{C}_{2} \rightarrow \mathrm{C}_{2}-\mathrm{C}_{1}, \mathrm{C}_{3} \rightarrow \mathrm{C}_{3}-\mathrm{C}_{1},$ we get

$\Delta  = \operatorname{abc} \left( {1 + \frac{1}{a} + \frac{1}{b} + \frac{1}{c}} \right)\left| {\begin{array}{*{20}{c}}
  1&0&0 \\ 
  {\frac{1}{b}}&1&0 \\ 
  {\frac{1}{c}}&0&1 
\end{array}} \right|$

$ = abc\left( {1 + \frac{1}{a} + \frac{1}{b} + \frac{1}{c}} \right)[1(1 – 0)]$

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

$ = abc + bc + ca + ab = {\text{R}}.{\text{H}}.{\text{S}}$