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Question
Evaluate the following determinant:
\[\begin{vmatrix}\cos 15^\circ & \sin 15^\circ \\ \sin 75^\circ & \cos 75^\circ\end{vmatrix}\]
Solution
\[∆ = \cos15^\circ\cos75^\circ - \sin15^\circ\sin75^\circ\]
\[ = \cos15^\circ\cos75^\circ - \sin(90^\circ - 75^\circ)\sin(90^\circ - 15^\circ) \left[ \because \sin\left( 90^\circ - \theta \right) = \cos\theta \right]\]
\[ = \cos15^\circ\cos75^\circ - \cos75^\circ\cos15^\circ\]
\[ = \cos15^\circ\cos75^\circ - \cos15^\circ\cos75^\circ = 0\]
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