Evaluate : ` |{:(1, cosec ^(2) theta, cot ^(2) theta),(-1 , cot ^(2) theta, cosec ^(2) theta),(2,(9)/(2), (5)/(2) ):}|` a) -1 b) 1 c) 0 d) None of these

Options

A. `-1`
B. 1
C. 0
D. None of these

Answer

To evaluate the determinant

\[ D = \begin{vmatrix} 1 & \csc^2 \theta & \cot^2 \theta \\ -1 & \cot^2 \theta & \csc^2 \theta \\ 2 & \frac{9}{2} & \frac{5}{2} \end{vmatrix} \]

we will use the method of cofactor expansion along the first column.

### Step 1: Expand the determinant using the first column

Using the first column, we have:

\[ D = 1 \cdot \begin{vmatrix} \cot^2 \theta & \csc^2 \theta \\ \frac{9}{2} & \frac{5}{2} \end{vmatrix} - (-1) \cdot \begin{vmatrix} \csc^2 \theta & \cot^2 \theta \\ \frac{9}{2} & \frac{5}{2} \end{vmatrix} + 2 \cdot \begin{vmatrix} \csc^2 \theta & \cot^2 \theta \\ \cot^2 \theta & \csc^2 \theta \end{vmatrix} \]

### Step 2: Calculate the 2x2 determinants

1. For the first determinant:

\[ \begin{vmatrix} \cot^2 \theta & \csc^2 \theta \\ \frac{9}{2} & \frac{5}{2} \end{vmatrix} = \cot^2 \theta \cdot \frac{5}{2} - \csc^2 \theta \cdot \frac{9}{2} = \frac{5}{2} \cot^2 \theta - \frac{9}{2} \csc^2 \theta \]

2. For the second determinant:

\[ \begin{vmatrix} \csc^2 \theta & \cot^2 \theta \\ \frac{9}{2} & \frac{5}{2} \end{vmatrix} = \csc^2 \theta \cdot \frac{5}{2} - \cot^2 \theta \cdot \frac{9}{2} = \frac{5}{2} \csc^2 \theta - \frac{9}{2} \cot^2 \theta \]

3. For the third determinant:

\[ \begin{vmatrix} \csc^2 \theta & \cot^2 \theta \\ \cot^2 \theta & \csc^2 \theta \end{vmatrix} = \csc^4 \theta - \cot^4 \theta \]

### Step 3: Substitute back into the determinant expression

Now substituting these back into our expression for \(D\):

\[ D = 1 \cdot \left( \frac{5}{2} \cot^2 \theta - \frac{9}{2} \csc^2 \theta \right) + \left( \frac{5}{2} \csc^2 \theta - \frac{9}{2} \cot^2 \theta \right) + 2 \cdot \left( \csc^4 \theta - \cot^4 \theta \right) \]

### Step 4: Combine like terms

Combining all the terms:

\[ D = \left( \frac{5}{2} \cot^2 \theta - \frac{9}{2} \csc^2 \theta + \frac{5}{2} \csc^2 \theta - \frac{9}{2} \cot^2 \theta + 2(\csc^4 \theta - \cot^4 \theta) \right) \]

### Step 5: Simplify the expression

This simplifies to:

\[ D = \left( \frac{5}{2} - \frac{9}{2} \right) \cot^2 \theta + \left( \frac{5}{2} - \frac{9}{2} \right) \csc^2 \theta + 2(\csc^4 \theta - \cot^4 \theta) \]

\[ D = -2 \cot^2 \theta - 2 \csc^2 \theta + 2(\csc^4 \theta - \cot^4 \theta) \]

### Step 6: Factor out common terms

Factoring out the common term:

\[ D = 2 \left( \csc^4 \theta - \cot^4 \theta - \cot^2 \theta - \csc^2 \theta \right) \]

### Step 7: Use the identity \(\csc^2 \theta - \cot^2 \theta = 1\)

Using the identity \(\csc^2 \theta - \cot^2 \theta = 1\):

\[ D = 2 \left( (\csc^2 \theta - \cot^2 \theta)(\csc^2 \theta + \cot^2 \theta) - \cot^2 \theta - \csc^2 \theta \right) \]
\[ D = 2 \left( 1 \cdot (\csc^2 \theta + \cot^2 \theta) - \cot^2 \theta - \csc^2 \theta \right) \]
\[ D = 2 \left( 0 \right) = 0 \]

### Final Answer

Thus, the value of the determinant is

\[ \boxed{0} \]