The minimum value of determinant `Delta=|{:(1,cos theta,1),(-cos theta , 1,cos theta),(-1,-cos theta, 2):}| AA theta in R` is :

Answer

To find the minimum value of the determinant

\[ \Delta = \begin{vmatrix} 1 & \cos \theta & 1 \\ -\cos \theta & 1 & \cos \theta \\ -1 & -\cos \theta & 2 \end{vmatrix} \]

we will calculate the determinant step by step.

### Step 1: Calculate the Determinant

Using the formula for the determinant of a 3x3 matrix:

\[ \Delta = a(ei - fh) - b(di - fg) + c(dh - eg) \]

where the matrix is

\[ \begin{pmatrix} a & b & c \\ d & e & f \\ g & h & i \end{pmatrix} \]

we can assign:

- \( a = 1, b = \cos \theta, c = 1 \)
- \( d = -\cos \theta, e = 1, f = \cos \theta \)
- \( g = -1, h = -\cos \theta, i = 2 \)

Now substituting into the determinant formula:

\[ \Delta = 1 \cdot (1 \cdot 2 - \cos \theta \cdot (-\cos \theta)) - \cos \theta \cdot (-\cos \theta \cdot 2 - \cos \theta \cdot (-1)) + 1 \cdot (-\cos \theta \cdot (-\cos \theta) - 1 \cdot 1) \]

### Step 2: Simplify Each Term

Calculating each term:

1. **First term**:
\[ 1 \cdot (2 + \cos^2 \theta) = 2 + \cos^2 \theta \]

2. **Second term**:
\[ -\cos \theta \cdot (-2 \cos \theta + \cos \theta) = -\cos \theta \cdot (-\cos \theta) = \cos^2 \theta \]

3. **Third term**:
\[ 1 \cdot (\cos^2 \theta - 1) = \cos^2 \theta - 1 \]

### Step 3: Combine All Terms

Now, we combine all the terms:

\[ \Delta = (2 + \cos^2 \theta) + \cos^2 \theta + (\cos^2 \theta - 1) \]

This simplifies to:

\[ \Delta = 2 + 3\cos^2 \theta - 1 = 1 + 3\cos^2 \theta \]

### Step 4: Find the Minimum Value

The term \(3\cos^2 \theta\) reaches its minimum value when \(\cos^2 \theta = 0\) (since \(\cos^2 \theta\) is always non-negative).

Thus, the minimum value of \(\Delta\) is:

\[ \Delta_{\text{min}} = 1 + 3 \cdot 0 = 1 \]

### Final Answer

The minimum value of the determinant \(\Delta\) is:

\[ \boxed{1} \]