Question

Without expanding, show that the value of the following determinant is zero:

\[\begin{vmatrix}\sin^2 A & \cot A & 1 \\ \sin^2 B & \cot B & 1 \\ \sin^2 C & \cot C & 1\end{vmatrix}, where A, B, C \text{ are the angles of }∆ ABC .\]

Answer 1

\[\begin{vmatrix}\sin^2 A & \cot A & 1 \\ \sin^2 B & \cot B & 1 \\ \sin^2 C & \cot C & 1\end{vmatrix}\]
\[ = \begin{vmatrix}\sin^2 A - \sin^2 B & \cot A - \cot B & 0 \\ \sin^2 B & \cot B & 1 \\ \sin^2 C - \sin^2 B & \cot C - \cot B & 0\end{vmatrix} \left[ \text{ Applying } R_1 \to R_1 - R_2 \text{ and }R_3 \to R_3 - R_2 \right]\]
\[ = \begin{vmatrix}\sin\left( A + B \right)\sin\left( A - B \right) & \frac{\cos A\sin B - \cos B\sin A}{\sin A\sin B} & 0 \\ \sin^2 B & \cot B & 1 \\ \sin\left( C + B \right)\sin\left( C - B \right) & \frac{\cos C\sin B - \cos B\sin C}{\sin B\sin C} & 0\end{vmatrix}\]
\[ = \begin{vmatrix}\sin\left( \pi - C \right)\sin\left( A - B \right) & \frac{- \sin\left( A - B \right)}{\sin A\sin B} & 0 \\ \sin^2 B & cot B & 1 \\ \sin\left( \pi - A \right)\sin\left( C - B \right) & \frac{- \sin\left( C - B \right)}{\sin B\sin C} & 0\end{vmatrix} \left[ \because A + B + C = \pi \right]\]
\[ = \begin{vmatrix}\sin C\sin\left( A - B \right) & \frac{- \sin\left( A - B \right)}{\sin A\sin B} & 0 \\ \sin^2 B & \frac{\cos B}{\sin B} & 1 \\ \sin A\sin\left( C - B \right) & \frac{- \sin\left( C - B \right)}{\sin B\sin C} & 0\end{vmatrix}\]
\[ = \frac{\sin\left( A - B \right)\sin\left( C - B \right)}{\sin B}\begin{vmatrix}\sin C & \frac{- 1}{\sin A} & 0 \\ \sin^2 B & \cos B & 1 \\ \sin A & \frac{- 1}{\sin C} & 0\end{vmatrix}\]
\[ = \frac{\sin\left( A - B \right)\sin\left( C - B \right)}{\sin B\sin A\sin C}\begin{vmatrix}\sin C\sin A & - 1 & 0 \\ \sin^2 B & \cos B & 1 \\ \sin A\sin C & - 1 & 0\end{vmatrix} \left[ \text{ Applying }R_1 \to \sin A R_1\text{  and }R_3 \to \sin C R_3 \right]\]
\[ = \frac{\sin\left( A - B \right)\sin\left( C - B \right)}{\sin B\sin A\sin C}\begin{vmatrix}0 & 0 & 0 \\ \sin^2 B & \cos B & 1 \\ \sin A\sin C & - 1 & 0\end{vmatrix} \left[ \text{ Applying }R_1 \to R_1 - R_3 \right]\]
\[ = 0\]