Question

The value of the determinant \[ \begin{vmatrix} \cos 75^\circ & \sin 75^\circ \\ \sin 15^\circ & \cos 15^\circ \end{vmatrix} \] is:

• A. \(1\)
• B. zero
• C. \(\dfrac{1}{2}\)
• D. \(\dfrac{\sqrt{3}}{2}\)

Hint:

Always check for opportunities to apply trigonometric identities like \(\cos(A + B)\) when evaluating determinants involving trigonometric terms. It simplifies the calculation quickly.

Answer 1

The Correct Option is B

We are given a \(2 \times 2\) determinant: \[ \begin{vmatrix} \cos 75^\circ & \sin 75^\circ
\sin 15^\circ & \cos 15^\circ \end{vmatrix} \] We apply the formula for the determinant of a \(2 \times 2\) matrix: \[ \text{Determinant} = (\cos 75^\circ)(\cos 15^\circ) - (\sin 75^\circ)(\sin 15^\circ) \] Using the trigonometric identity: \[ \cos A \cos B - \sin A \sin B = \cos(A + B) \] Here, \(A = 75^\circ\), \(B = 15^\circ\): \[ \cos 75^\circ \cos 15^\circ - \sin 75^\circ \sin 15^\circ = \cos(75^\circ + 15^\circ) = \cos(90^\circ) = 0 \] \[ \boxed{\text{Determinant} = 0} \]