Without expanding, find the value of `|(cosec^(2)theta,cot^(2)theta,1),(cot^(2)theta,cosec^(2)theta,-1),(42,40,2)|`

Answer

To find the value of the determinant

\[ D = \begin{vmatrix} \csc^2 \theta & \cot^2 \theta & 1 \\ \cot^2 \theta & \csc^2 \theta & -1 \\ 42 & 40 & 2 \end{vmatrix} \]

we will use properties of determinants without expanding it.

### Step 1: Modify the Determinant
We can perform column operations on the determinant. Specifically, we will subtract the second column from the first column. This gives us:

\[ D = \begin{vmatrix} \csc^2 \theta - \cot^2 \theta & \cot^2 \theta & 1 \\ \cot^2 \theta - \csc^2 \theta & \csc^2 \theta & -1 \\ 42 - 40 & 40 & 2 \end{vmatrix} \]

### Step 2: Simplify the Entries
Now, simplify the entries in the first column:

- The first row becomes \(\csc^2 \theta - \cot^2 \theta\)
- The second row becomes \(\cot^2 \theta - \csc^2 \theta\)
- The third row becomes \(2\)

Thus, the determinant now looks like:

\[ D = \begin{vmatrix} \csc^2 \theta - \cot^2 \theta & \cot^2 \theta & 1 \\ \cot^2 \theta - \csc^2 \theta & \csc^2 \theta & -1 \\ 2 & 40 & 2 \end{vmatrix} \]

### Step 3: Observe the Columns
Notice that the first column can be rewritten as:

\[ \begin{vmatrix} \csc^2 \theta - \cot^2 \theta & \cot^2 \theta & 1 \\ \cot^2 \theta - \csc^2 \theta & \csc^2 \theta & -1 \\ 2 & 40 & 2 \end{vmatrix} \]

### Step 4: Check for Equal Columns
Now, observe that if we denote the first column as \(C_1\) and the second column as \(C_2\), we see that:

- \(C_1 = \begin{pmatrix} \csc^2 \theta - \cot^2 \theta \\ \cot^2 \theta - \csc^2 \theta \\ 2 \end{pmatrix}\)
- \(C_2 = \begin{pmatrix} \cot^2 \theta \\ \csc^2 \theta \\ 40 \end{pmatrix}\)

However, we can see that the first and second rows of the first column are negatives of each other. This means that the first and second rows of the determinant are linearly dependent.

### Step 5: Conclusion
Since we have two rows (or columns) that are linearly dependent, the value of the determinant must be zero.

Thus, we conclude that:

\[ D = 0 \]

### Final Answer
The value of the determinant is \(0\).

---