The determinant `D=|{:(cos(alpha+beta),-sin(alpha+beta),cos2beta),(sinalpha,cosalpha,sinbeta),(-cosalpha,sinalpha,cosbeta):}|` is independent of :-

Options

A. `alpha`
B. `beta`
C. `alpha" and "beta`
D. Neihter `alpha"nor"beta`

Answer

To solve the determinant \( D = \begin{vmatrix} \cos(\alpha + \beta) & -\sin(\alpha + \beta) & \cos(2\beta) \\ \sin \alpha & \cos \alpha & \sin \beta \\ -\cos \alpha & \sin \alpha & \cos \beta \end{vmatrix} \) and determine what it is independent of, we will follow these steps:

### Step 1: Expand the Determinant
We will expand the determinant along the first row (R1). The formula for the determinant of a 3x3 matrix is given by:
\[ D = a(ei - fh) - b(di - fg) + c(dh - eg) \]
where \( a, b, c \) are the elements of the first row, and \( d, e, f, g, h, i \) are the elements of the second and third rows.

For our determinant:
- \( a = \cos(\alpha + \beta) \)
- \( b = -\sin(\alpha + \beta) \)
- \( c = \cos(2\beta) \)

The remaining elements are:
- \( d = \sin \alpha \)
- \( e = \cos \alpha \)
- \( f = \sin \beta \)
- \( g = -\cos \alpha \)
- \( h = \sin \alpha \)
- \( i = \cos \beta \)

### Step 2: Calculate Each Component
1. **First Component**:
\[ D_1 = \cos(\alpha + \beta) \left( \cos \alpha \cdot \cos \beta - \sin \alpha \cdot \sin \beta \right) = \cos(\alpha + \beta) \cdot \cos(\alpha + \beta) = \cos^2(\alpha + \beta) \]

2. **Second Component**:
\[ D_2 = -(-\sin(\alpha + \beta)) \left( \sin \alpha \cdot \cos \beta - \sin \beta \cdot (-\cos \alpha) \right) = \sin(\alpha + \beta) \left( \sin \alpha \cdot \cos \beta + \sin \beta \cdot \cos \alpha \right) = \sin(\alpha + \beta) \cdot \sin(\alpha + \beta) = \sin^2(\alpha + \beta) \]

3. **Third Component**:
\[ D_3 = \cos(2\beta) \left( \sin \alpha \cdot \sin \alpha - \cos \alpha \cdot (-\cos \alpha) \right) = \cos(2\beta) \left( \sin^2 \alpha + \cos^2 \alpha \right) = \cos(2\beta) \cdot 1 = \cos(2\beta) \]

### Step 3: Combine the Components
Putting it all together, we have:
\[ D = \cos^2(\alpha + \beta) + \sin^2(\alpha + \beta) + \cos(2\beta) \]
Using the identity \( \cos^2 x + \sin^2 x = 1 \):
\[ D = 1 + \cos(2\beta) \]

### Step 4: Analyze Independence
The expression \( D = 1 + \cos(2\beta) \) shows that \( D \) depends only on \( \beta \) and is independent of \( \alpha \).

### Final Answer
Thus, the determinant \( D \) is independent of \( \alpha \).

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