Question:
One of the values of \( x \) for which \( \begin{vmatrix} \cos x & \sin x \\ -\cos x & \sin x \end{vmatrix} = 1 \) is
One of the values of \( x \) for which \( \begin{vmatrix} \cos x & \sin x \\ -\cos x & \sin x \end{vmatrix} = 1 \) is
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Tip 1: If you forget the \( \sin 2x \) identity, just substitute the options into the expanded expression \( 2\sin x \cos x \).
Tip 2: At \( x = \pi/4 \), \( \sin x = \cos x = 1/\sqrt{2} \), so \( 2(1/\sqrt{2})(1/\sqrt{2}) = 1 \).
Tip 2: At \( x = \pi/4 \), \( \sin x = \cos x = 1/\sqrt{2} \), so \( 2(1/\sqrt{2})(1/\sqrt{2}) = 1 \).
Updated On: Sep 10, 2026
- \( 0 \)
- \( \frac{\pi}{4} \)
- \( \frac{\pi}{3} \)
- \( \frac{\pi}{2} \)
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The Correct Option is B
Solution and Explanation
Concept:
• The determinant of a \( 2 \times 2 \) matrix \( \begin{vmatrix} a & b c & d \end{vmatrix} \) is \( ad - bc \).
• Trigonometric Identity: \( 2 \sin \theta \cos \theta = \sin 2\theta \).
Step 1: Expand the given determinant
\[ \begin{vmatrix} \cos x & \sin x \\ -\cos x & \sin x \end{vmatrix} = (\cos x)(\sin x) - (\sin x)(-\cos x) \] \[ = \cos x \sin x + \sin x \cos x \] \[ = 2 \sin x \cos x \]
Step 2: Equate to the given value
Using the identity \( 2 \sin x \cos x = \sin 2x \): \[ \sin 2x = 1 \]
Step 3: Solve for \( x \)
For \( \sin \theta = 1 \), the general solution is \( \theta = n\pi + (-1)^n \frac{\pi}{2} \). For the principal value: \[ 2x = \frac{\pi}{2} \] \[ x = \frac{\pi}{4} \]
• The determinant of a \( 2 \times 2 \) matrix \( \begin{vmatrix} a & b c & d \end{vmatrix} \) is \( ad - bc \).
• Trigonometric Identity: \( 2 \sin \theta \cos \theta = \sin 2\theta \).
Step 1: Expand the given determinant
\[ \begin{vmatrix} \cos x & \sin x \\ -\cos x & \sin x \end{vmatrix} = (\cos x)(\sin x) - (\sin x)(-\cos x) \] \[ = \cos x \sin x + \sin x \cos x \] \[ = 2 \sin x \cos x \]
Step 2: Equate to the given value
Using the identity \( 2 \sin x \cos x = \sin 2x \): \[ \sin 2x = 1 \]
Step 3: Solve for \( x \)
For \( \sin \theta = 1 \), the general solution is \( \theta = n\pi + (-1)^n \frac{\pi}{2} \). For the principal value: \[ 2x = \frac{\pi}{2} \] \[ x = \frac{\pi}{4} \]
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