If `|(1+sin^2 theta,sin^2 theta,sin^2 theta),(cos^2 theta,1+cos^2 theta,cos^2 theta),(4sin 4 theta,4sin4theta,1+4sin4theta)|=0,` then ...`sin 4theta` equal to ....

Options

A. `1//2`
B. 1
C. `-1//2`
D. `-1`

Explanation

To solve the problem, we need to evaluate the determinant of the given matrix and set it equal to zero. The matrix is:

\[ \begin{pmatrix} 1 + \sin^2 \theta & \sin^2 \theta & \sin^2 \theta \\ \cos^2 \theta & 1 + \cos^2 \theta & \cos^2 \theta \\ 4 \sin 4\theta & 4 \sin 4\theta & 1 + 4 \sin 4\theta \end{pmatrix} \]

### Step 1: Calculate the determinant

We can use the property of determinants that allows us to perform column operations without changing the value of the determinant. We will perform the following operations:

1. \( C_1 \leftarrow C_1 - C_2 \)
2. \( C_2 \leftarrow C_2 - C_3 \)

This gives us:

\[ \begin{pmatrix} 1 + \sin^2 \theta - \sin^2 \theta & \sin^2 \theta - \sin^2 \theta & \sin^2 \theta \\ \cos^2 \theta - \cos^2 \theta & 1 + \cos^2 \theta - \cos^2 \theta & \cos^2 \theta \\ 4 \sin 4\theta - 4 \sin 4\theta & 4 \sin 4\theta - (1 + 4 \sin 4\theta) & 1 + 4 \sin 4\theta \end{pmatrix} \]

This simplifies to:

\[ \begin{pmatrix} 1 & 0 & \sin^2 \theta \\ 0 & 1 & \cos^2 \theta \\ 0 & -1 & 1 + 4 \sin 4\theta \end{pmatrix} \]

### Step 2: Calculate the new determinant

Now, we can calculate the determinant of the new matrix:

\[ \text{Det} = 1 \cdot \begin{vmatrix} 1 & \sin^2 \theta \\ -1 & 1 + 4 \sin 4\theta \end{vmatrix} \]

Calculating this 2x2 determinant:

\[ = 1 \cdot (1 \cdot (1 + 4 \sin 4\theta) - (-1) \cdot \sin^2 \theta) = 1 + 4 \sin 4\theta + \sin^2 \theta \]

### Step 3: Set the determinant to zero

Now, we set the determinant equal to zero:

\[ 1 + 4 \sin 4\theta + \sin^2 \theta = 0 \]

### Step 4: Solve for \( \sin 4\theta \)

Rearranging the equation gives:

\[ 4 \sin 4\theta = -1 - \sin^2 \theta \]

Now, we can express \( \sin 4\theta \):

\[ \sin 4\theta = \frac{-1 - \sin^2 \theta}{4} \]

### Step 5: Find the value of \( \sin 4\theta \)

To find a specific value, we can assume \( \sin^2 \theta = 0 \) (which corresponds to \( \theta = 0 \) or \( \theta = \pi \)):

\[ \sin 4\theta = \frac{-1 - 0}{4} = -\frac{1}{4} \]

Thus, the final answer is:

\[ \sin 4\theta = -\frac{1}{4} \]