The maximum value of determinant `|{:(cos(alpha+beta),sin alpha,-cos alpha),(-sin(alpha+beta),cos alpha, sin alpha),(cos 2beta,sin beta, cos beta):}|` is ___________

Answer

To find the maximum value of the determinant

\[ D = \begin{vmatrix} \cos(\alpha + \beta) & \sin \alpha & -\cos \alpha \\ -\sin(\alpha + \beta) & \cos \alpha & \sin \alpha \\ \cos 2\beta & \sin \beta & \cos \beta \end{vmatrix} \]

we will follow these steps:

### Step 1: Expand the determinant

Using the property of determinants, we can expand \(D\) as follows:

\[ D = \cos(\alpha + \beta) \begin{vmatrix} \cos \alpha & \sin \alpha \\ \sin \beta & \cos \beta \end{vmatrix} + \sin \alpha \begin{vmatrix} -\sin(\alpha + \beta) & \sin \alpha \\ \cos 2\beta & \cos \beta \end{vmatrix} - \cos \alpha \begin{vmatrix} -\sin(\alpha + \beta) & \cos \alpha \\ \cos 2\beta & \sin \beta \end{vmatrix} \]

### Step 2: Calculate the 2x2 determinants

1. For the first determinant:

\[ \begin{vmatrix} \cos \alpha & \sin \alpha \\ \sin \beta & \cos \beta \end{vmatrix} = \cos \alpha \cos \beta - \sin \alpha \sin \beta = \cos(\alpha + \beta) \]

2. For the second determinant:

\[ \begin{vmatrix} -\sin(\alpha + \beta) & \sin \alpha \\ \cos 2\beta & \cos \beta \end{vmatrix} = -\sin(\alpha + \beta) \cos \beta - \sin \alpha \cos 2\beta \]

3. For the third determinant:

\[ \begin{vmatrix} -\sin(\alpha + \beta) & \cos \alpha \\ \cos 2\beta & \sin \beta \end{vmatrix} = -\sin(\alpha + \beta) \sin \beta + \cos \alpha \cos 2\beta \]

### Step 3: Substitute back into the determinant

Now substituting these back into \(D\):

\[ D = \cos(\alpha + \beta) \cdot \cos(\alpha + \beta) + \sin \alpha (-\sin(\alpha + \beta) \cos \beta - \sin \alpha \cos 2\beta) - \cos \alpha (-\sin(\alpha + \beta) \sin \beta + \cos \alpha \cos 2\beta) \]

### Step 4: Simplify the expression

After simplification, we will find that:

\[ D = 2 \cos^2 \beta \]

### Step 5: Find the maximum value

The maximum value of \(\cos^2 \beta\) is 1 (since \(\cos^2 \beta\) ranges from 0 to 1). Therefore, the maximum value of \(D\) is:

\[ D_{\text{max}} = 2 \cdot 1 = 2 \]

### Final Answer

Thus, the maximum value of the determinant is:

\[ \boxed{2} \]