Explanation

(a)
Given, $\left| \begin{matrix}
a & \cot\frac{A}{2} & \lambda \\
b & \cot\frac{B}{2} & \mu \\
c & \cot\frac{C}{2} & \gamma
\end{matrix} \right| = 0$
$\Rightarrow \ \left| \begin{matrix}
a & \frac{s(s - a)}{\mathrm{\Delta}} & \lambda \\
b & \frac{s(s - b)}{\mathrm{\Delta}} & \mu \\
c & \frac{s(s - c)}{\mathrm{\Delta}} & \gamma
\end{matrix} \right| = 0$
$\left\lbrack \because\cot{\frac{A}{2} = \frac{s(s - a)}{\sqrt{(s - a)(s - b)(s - c)}} = \frac{s(s - a)}{\mathrm{\Delta}}} \right\rbrack$
$\Rightarrow \frac{1}{r}\left| \begin{matrix}
a & s - a & \lambda \\
b & s - b & \mu \\
c & s - c & \gamma
\end{matrix} \right| = 0\ \ \ where\ r = \frac{\mathrm{\Delta}}{s}$
Applying $C_{2} \rightarrow C_{2} + C_{1}$
$\Rightarrow \frac{1}{r}\left| \begin{matrix}
a & s & \lambda \\
b & s & \mu \\
c & s & \gamma
\end{matrix} \right| = 0$
$\Rightarrow \frac{\mathrm{\Delta}}{r^{2}}\left| \begin{matrix}
a & 1 & \lambda \\
b & 1 & \mu \\
c & 1 & \gamma
\end{matrix} \right| = 0$
Applying $R_{1} \rightarrow {R_{1} - R}_{2,\ }R_{2} \rightarrow R_{2} - R_{3}$
$\Rightarrow \ \frac{\mathrm{\Delta}}{r^{2}}\left| \begin{matrix}
a - b & 0 & \lambda - \mu \\
b - c & 0 & \mu - \gamma \\
c & 1 & \gamma
\end{matrix} \right| = 0$
$\Rightarrow \ \frac{\mathrm{\Delta}}{r^{2}}\left\lbrack (b - c)(\lambda - \mu) - (\mu - \gamma)(a - b) \right\rbrack = 0$
$\Rightarrow b(\lambda - \mu) - c(\lambda - \mu) - a(\mu - \gamma) + b(\mu - \gamma) = 0$
$\Rightarrow \ - a(\mu - \gamma) + b(\lambda - \mu + \mu - \gamma) - c(\lambda - \mu) = 0$
$\Rightarrow \ - a(\mu - \gamma) + b(\lambda - \gamma) - c(\lambda - \mu) = 0$
$\Rightarrow a(\mu - \gamma) + b(\gamma - \lambda) + c(\lambda - \mu) = 0$