Question

The value of \(\begin{vmatrix}     \sin^2 14 \degree & \sin^2 66\degree & \tan 135\degree \\     \sin^2 66\degree & \tan 135\degree &     \sin^2 14 \degree \\     \tan 135\degree & \sin^2 14 \degree & \sin^2 66\degree \end{vmatrix}\)

• A. 1
• B. -1
• C. 2
• D. 0

Hint:

When you encounter a matrix with rows or columns that exhibit symmetry, it's often a good idea to check if the rows or columns are linearly dependent. If they are, the determinant of the matrix will be zero.

Answer 1

The Correct Option is D

The correct answer is: (D): 0

We are tasked with finding the value of the following determinant:

\(\begin{vmatrix} \sin^2 14^\circ & \sin^2 66^\circ & \tan 135^\circ \\ \sin^2 66^\circ & \tan 135^\circ & \sin^2 14^\circ \\ \tan 135^\circ & \sin^2 14^\circ & \sin^2 66^\circ \end{vmatrix}\)

Step 1: Simplify the trigonometric values

First, recall that \( \tan 135^\circ = -1 \), because the tangent of 135° is -1 (since \( \tan(180^\circ - x) = -\tan x \)). Therefore, we can replace all instances of \( \tan 135^\circ \) in the matrix with -1:

\(\begin{vmatrix} \sin^2 14^\circ & \sin^2 66^\circ & -1 \\ \sin^2 66^\circ & -1 & \sin^2 14^\circ \\ -1 & \sin^2 14^\circ & \sin^2 66^\circ \end{vmatrix}\)

Step 2: Analyze the structure of the matrix

Now, observe that this is a symmetric matrix where the elements in the first and second rows are switched between the columns. This symmetry indicates that the rows are linearly dependent, meaning one row can be expressed as a linear combination of the others.

Step 3: Conclusion

The determinant of a matrix with linearly dependent rows is 0. Therefore, the value of the determinant is 0.

Conclusion:
The value of the determinant is 0, so the correct answer is (D): 0.