The value of the determinant
`Delta = |(cos (alpha + beta),- sin (alpha + beta),cos 2 beta),(sin alpha,cos alpha,sin beta),(- cos alpha,sin alpha,- cos beta)|`, is

Options

A. `cos^(2) alpha`
B. `sin^(2) alpha`
C. `sin (alpha - beta)`
D. 0

Explanation

To find the value of the determinant

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

we will apply properties of determinants to simplify the calculation.

### Step 1: Apply Row Operations
We can simplify the first row by adding appropriate multiples of the second and third rows to it. Specifically, we will perform the following operations:
- Replace Row 1 with Row 1 + \(\sin(\beta) \cdot \text{Row 2}\) + \(\cos(\beta) \cdot \text{Row 3}\).

### Step 2: Calculate New Elements of Row 1
1. First element of Row 1:
\[ \cos(\alpha + \beta) + \sin(\beta) \cdot \sin(\alpha) + \cos(\beta) \cdot (-\cos(\alpha)) \]
This simplifies to:
\[ \cos(\alpha + \beta) + \sin(\beta) \sin(\alpha) - \cos(\beta) \cos(\alpha) \]
Using the cosine addition formula, we have:
\[ \cos(\alpha + \beta) - \cos(\alpha + \beta) = 0 \]

2. Second element of Row 1:
\[ -\sin(\alpha + \beta) + \sin(\beta) \cdot \cos(\alpha) + \cos(\beta) \cdot \sin(\alpha) \]
This simplifies to:
\[ -\sin(\alpha + \beta) + \sin(\beta) \cos(\alpha) + \cos(\beta) \sin(\alpha) \]
Using the sine addition formula, we have:
\[ -\sin(\alpha + \beta) + \sin(\alpha + \beta) = 0 \]

3. Third element of Row 1:
\[ \cos(2\beta) + \sin(\beta) \cdot \sin(\beta) + \cos(\beta) \cdot (-\cos(\beta)) \]
This simplifies to:
\[ \cos(2\beta) + \sin^2(\beta) - \cos^2(\beta) \]
Using the identity \(\cos(2\beta) = \cos^2(\beta) - \sin^2(\beta)\), we find:
\[ \cos(2\beta) - \cos(2\beta) = 0 \]

### Step 3: Form the New Determinant
After performing the row operations, Row 1 becomes:
\[ \begin{pmatrix} 0 & 0 & 0 \end{pmatrix} \]
Thus, the determinant can be rewritten as:
\[ \Delta = \begin{vmatrix} 0 & 0 & 0 \\ \sin(\alpha) & \cos(\alpha) & \sin(\beta) \\ -\cos(\alpha) & \sin(\alpha) & -\cos(\beta) \end{vmatrix} \]

### Step 4: Evaluate the Determinant
Since the first row consists entirely of zeros, the value of the determinant is:
\[ \Delta = 0 \]

### Conclusion
The value of the determinant \(\Delta\) is \(0\).

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