Question

Evaluate: \[\begin{vmatrix}\cos 15^\circ & \sin 15^\circ \\ \sin 75^\circ & \cos 75^\circ\end{vmatrix}\]

Answer 1

\[\begin{vmatrix}\cos 15^\circ & \sin 15^\circ \\ \sin 75^\circ & \cos 75^\circ\end{vmatrix}\]
\[ = \cos 15^\circ \cos 75^\circ - \sin 15^\circ \sin 75^\circ\] 
\[ = \cos (15^\circ + 75^\circ) \left[ \because \cos A \cos B - \sin A \sin B = \cos (A + B) \right]\] \[ = \cos 90^\circ\] 
\[ = 0\] 
\[ \Rightarrow \begin{vmatrix}\cos 15^\circ & \sin 15^\circ \\ \sin 75^\circ & \cos 75^\circ\end{vmatrix}\] = 0\]