If A, B and C are angles of triangle, then the value of $\left|\begin{array}{ccc}{\sin }^2\mathrm{A}& \cot \mathrm{A}& 1\\ {\sin }^2\mathrm{B}& \cot \mathrm{B}& 1\\ {\sin }^2\mathrm{C}& \cot \mathrm{C}& 1\end{array}\right|$ is

Applying ${\mathrm{R}}_2\to {\mathrm{R}}_2-{\mathrm{R}}_1\text{ and }{\mathrm{R}}_3\to {\mathrm{R}}_3-{\mathrm{R}}_1$, we get

$\mathrm{\Delta }=\left|\begin{array}{ccc}{\sin }^2\mathrm{A}& \cot \mathrm{A}& 1\\ \sin (\mathrm{B}+\mathrm{A})\sin (\mathrm{B}-\mathrm{A})& \frac{\sin (\mathrm{A}-\mathrm{B})}{\sin \mathrm{Asin}\mathrm{B}}& 0\\ \sin (\mathrm{C}+\mathrm{A})\sin (\mathrm{C}-\mathrm{A})& \frac{\sin (\mathrm{A}-\mathrm{C})}{\sin \mathrm{Asin}\mathrm{C}}& 0\end{array}\right|$             $\left[\because \cot \mathrm{\alpha }-\cot \mathrm{\beta }=\frac{\sin (\mathrm{\beta }-\mathrm{\alpha })}{\sin \mathrm{\alpha sin}\mathrm{\beta }}\right]$

Expanding along C₃, we get

$\begin{array}{c}\mathrm{\Delta }=\frac{\sin (\mathrm{A}-\mathrm{B})\sin (\mathrm{A}-\mathrm{C})}{\sin \mathrm{A}}\left[-\frac{\sin (\mathrm{B}+\mathrm{A})}{\sin \mathrm{C}}+\frac{\sin (\mathrm{C}+\mathrm{A})}{\sin \mathrm{B}}\right]\\ =\frac{\sin (\mathrm{A}-\mathrm{B})\sin (\mathrm{A}-\mathrm{C})}{\sin \mathrm{A}}\left[-\frac{\sin (\mathrm{\pi }-\mathrm{C})}{\sin \mathrm{C}}+\frac{\sin (\mathrm{\pi }-\mathrm{B})}{\sin \mathrm{B}}\right]\\ =\frac{\sin (\mathrm{A}-\mathrm{B})\sin (\mathrm{A}-\mathrm{C})}{\sin \mathrm{A}}\left[-\frac{\sin \mathrm{C}}{\sin \mathrm{C}}+\frac{\sin \mathrm{B}}{\sin \mathrm{B}}\right]=0\end{array}$