Question

The value of the determinant \[ \begin{vmatrix} \cos^2 54^\circ & \cos^2 36^\circ & \cot 135^\circ \\ \sin^2 53^\circ & \cot 135^\circ & \sin^2 37^\circ \\ \cot 135^\circ & \cos^2 25^\circ & \cos^2 65^\circ \end{vmatrix} \] is equal to

• A. \(-2\)
• B. \(-1\)
• C. \(0\)
• D. \(1\)
• E. \(2\)

Hint:

Use complementary angle identities to simplify trigonometric determinants.

Answer 1

The Correct Option is C

Concept: Use identities: \[ \sin^2\theta + \cos^2\theta = 1,\quad \cot 135^\circ = -1 \]

Step 1: Evaluate constants
\[ \cot 135^\circ = -1 \]

Step 2: Use complementary angles
\[ \cos^2 54^\circ = \sin^2 36^\circ \] \[ \sin^2 53^\circ = \cos^2 37^\circ \] \[ \cos^2 65^\circ = \sin^2 25^\circ \]

Step 3: Substitute
Matrix becomes symmetric in structure.

Step 4: Observe rows/columns relation
Rows become linearly dependent.

Step 5: Determinant
If rows dependent ⇒ determinant = 0 \[ \boxed{0} \]