If A,B,C be the angles of a triangle , then the value of `Delta=|(-1,cosC,cosB),(cosC,-1,cosA),(cosB,cosA,-1)|` is

Options

A. `cos A cos B cos C`
B. `sin A sinB sinC `
C. 0
D. none of these

Answer

To find the value of the determinant

\[ \Delta = \begin{vmatrix} -1 & \cos C & \cos B \\ \cos C & -1 & \cos A \\ \cos B & \cos A & -1 \end{vmatrix} \]

we will expand this determinant using the first row.

### Step 1: Expand the Determinant
Using the cofactor expansion along the first row, we have:

\[ \Delta = -1 \cdot \begin{vmatrix} -1 & \cos A \\ \cos A & -1 \end{vmatrix} - \cos C \cdot \begin{vmatrix} \cos C & \cos A \\ \cos B & -1 \end{vmatrix} + \cos B \cdot \begin{vmatrix} \cos C & -1 \\ \cos B & \cos A \end{vmatrix} \]

### Step 2: Calculate the 2x2 Determinants
Now we calculate each of the 2x2 determinants.

1. For the first determinant:
\[ \begin{vmatrix} -1 & \cos A \\ \cos A & -1 \end{vmatrix} = (-1)(-1) - (\cos A)(\cos A) = 1 - \cos^2 A = \sin^2 A \]

2. For the second determinant:
\[ \begin{vmatrix} \cos C & \cos A \\ \cos B & -1 \end{vmatrix} = (\cos C)(-1) - (\cos A)(\cos B) = -\cos C - \cos A \cos B \]

3. For the third determinant:
\[ \begin{vmatrix} \cos C & -1 \\ \cos B & \cos A \end{vmatrix} = (\cos C)(\cos A) - (-1)(\cos B) = \cos C \cos A + \cos B \]

### Step 3: Substitute Back into the Determinant
Now substituting these values back into the expression for \(\Delta\):

\[ \Delta = -1 \cdot \sin^2 A - \cos C \cdot (-\cos C - \cos A \cos B) + \cos B \cdot (\cos C \cos A + \cos B) \]

This simplifies to:

\[ \Delta = -\sin^2 A + \cos C(\cos C + \cos A \cos B) + \cos B(\cos C \cos A + \cos B) \]

### Step 4: Simplify Further
Now we can simplify further:

\[ \Delta = -\sin^2 A + \cos^2 C + \cos C \cos A \cos B + \cos B \cos C \cos A + \cos^2 B \]

### Step 5: Use the Identity \(\sin^2 A + \sin^2 B + \sin^2 C = 1\)
Using the identity for angles in a triangle, we can also express \(\sin^2 A\) in terms of \(\cos\):

\[ \Delta = \cos^2 A + \cos^2 B + \cos^2 C - 1 \]

### Final Result
Thus, the value of the determinant \(\Delta\) is:

\[ \Delta = 0 \]