The determinant
`|(sec^(2)theta, tan^(2)theta,1),(tan^(2)theta, sec^(2)theta, -1),(12,10,2)|` equals

Options

A. `2sin^(2)theta`
B. `12sec^(2)theta-10tan^(2)theta`
C. `12sec^(2)theta-10tan^(2)theta+5`
D. 0

Answer

To find the value of the determinant

\[ D = \begin{vmatrix} \sec^2 \theta & \tan^2 \theta & 1 \\ \tan^2 \theta & \sec^2 \theta & -1 \\ 12 & 10 & 2 \end{vmatrix} \]

we can follow these steps:

### Step 1: Identify the determinant
We start with the given determinant:

\[ D = \begin{vmatrix} \sec^2 \theta & \tan^2 \theta & 1 \\ \tan^2 \theta & \sec^2 \theta & -1 \\ 12 & 10 & 2 \end{vmatrix} \]

### Step 2: Use properties of determinants
We know that \( \tan^2 \theta + 1 = \sec^2 \theta \). This property will help us simplify the determinant.

### Step 3: Modify the second column
Let's add the third column to the second column:

\[ D = \begin{vmatrix} \sec^2 \theta & \tan^2 \theta + 1 & 1 \\ \tan^2 \theta & \sec^2 \theta - 1 & -1 \\ 12 & 10 + 2 & 2 \end{vmatrix} \]

This gives us:

\[ D = \begin{vmatrix} \sec^2 \theta & \sec^2 \theta & 1 \\ \tan^2 \theta & \tan^2 \theta & -1 \\ 12 & 12 & 2 \end{vmatrix} \]

### Step 4: Identify proportional columns
Now we can see that the first two columns are proportional (both are equal to each other).

### Step 5: Apply the property of determinants
According to the property of determinants, if two columns (or rows) of a determinant are proportional, then the value of the determinant is zero.

### Conclusion
Thus, we conclude that:

\[ D = 0 \]