If \[\left| {\begin{array}{*{20}{c}} a&{\cot \frac{A}{2}}&\lambda \\ b&{\cot \frac{B}{2}}&\mu \\ c&{\cot \frac{C}{2}}&\gamma \end{array}} \right| = 0,\]    where $$a, b, c, A, B,$$   and $$C$$ are elements of a triangle $$ABC$$  with usual meaning. Then, the value of a $$\left( {\mu - \gamma } \right) + b\left( {\gamma - \lambda } \right) + c\left( {\lambda - \mu } \right) = 0$$        is

Solution :
Given, \[\left| {\begin{array}{*{20}{c}} a&{\cot \frac{A}{2}}&\lambda \\ b&{\cot \frac{B}{2}}&\mu \\ c&{\cot \frac{C}{2}}&\gamma \end{array}} \right| = 0\]
\[\begin{array}{l} \Rightarrow \left| {\begin{array}{*{20}{c}} a&{\frac{{s\left( {s - a} \right)}}{\Delta }}&\lambda \\ b&{\frac{{s\left( {s - b} \right)}}{\Delta }}&\mu \\ c&{\frac{{s\left( {s - c} \right)}}{\Delta }}&\gamma \end{array}} \right| = 0\\ \Rightarrow \frac{1}{r}\left| {\begin{array}{*{20}{c}} a&{s - a}&\lambda \\ b&{s - b}&\mu \\ c&{s - c}&\gamma \end{array}} \right| = 0 \end{array}\]
Apply, $${C_2} \to {C_2} + {C_1}$$
\[\begin{array}{l} \Rightarrow \frac{1}{r}\left| {\begin{array}{*{20}{c}} a&s&\lambda \\ b&s&\mu \\ c&s&\gamma \end{array}} \right| = 0,\,\,{\rm{where}}\,\,r = \frac{\Delta }{s}\\ \Rightarrow \frac{\Delta }{{{r^2}}}\left| {\begin{array}{*{20}{c}} a&1&\lambda \\ b&1&\mu \\ c&1&\gamma \end{array}} \right| = 0 \end{array}\]
Apply, $${R_1} \to {R_1} - {R_2},{R_2} \to {R_2} - {R_3}$$
\[ \Rightarrow \frac{\Delta }{{{r^2}}}\left| {\begin{array}{*{20}{c}} {a - b}&0&{\lambda - \mu }\\ {b - c}&0&{\mu - \gamma }\\ c&1&\gamma \end{array}} \right| = 0\]
$$\eqalign{ & \Rightarrow \frac{\Delta }{{{r^2}}}\left[ {\left( {b - c} \right)\left( {\lambda - \mu } \right) - \left( {\mu - \gamma } \right)\left( {a - b} \right)} \right] = 0 \cr & \Rightarrow b\left( {\lambda - \mu } \right) - c\left( {\lambda - \mu } \right) - a\left( {\mu - \gamma } \right) + b\left( {\mu - \gamma } \right) = 0 \cr & \Rightarrow - a\left( {\mu - \gamma } \right) + b\left( {\lambda - \mu + \mu - \gamma } \right) - c\left( {\lambda - \mu } \right) = 0 \cr & \Rightarrow - a\left( {\mu - \gamma } \right) + b\left( {\lambda - \gamma } \right) - c\left( {\lambda - \mu } \right) = 0 \cr & \Rightarrow a\left( {\mu - \gamma } \right) + b\left( {\gamma - \lambda } \right) + c\left( {\lambda - \mu } \right) = 0 \cr} $$