Let `A and B` denote the statements
`A : cos alpha + cos beta + cos gamma =0`
`B : sin alpha + siin beta + sin gamma = 0`
If `cos(beta - gamma) + cos (gamma -alpha) + cos (alpha -beta) = - (3)/(2)`,
then

Options

A. `A` is true and `B` is false.
B. `A` is false and `B` is true.
C. Both `A and B` are true.
D. Both `A and B` are false.

Explanation

To solve the problem, we need to analyze the given equations and statements step by step.

### Step 1: Understand the Given Information
We are given two statements:
- Statement A: \( \cos \alpha + \cos \beta + \cos \gamma = 0 \)
- Statement B: \( \sin \alpha + \sin \beta + \sin \gamma = 0 \)

We also have the equation:
\[ \cos(\beta - \gamma) + \cos(\gamma - \alpha) + \cos(\alpha - \beta) = -\frac{3}{2} \]

### Step 2: Use the Cosine Addition Formula
We can use the cosine subtraction formula:
\[ \cos(x - y) = \cos x \cos y + \sin x \sin y \]
Applying this to our equation:
\[ \cos(\beta - \gamma) = \cos \beta \cos \gamma + \sin \beta \sin \gamma \]
\[ \cos(\gamma - \alpha) = \cos \gamma \cos \alpha + \sin \gamma \sin \alpha \]
\[ \cos(\alpha - \beta) = \cos \alpha \cos \beta + \sin \alpha \sin \beta \]

### Step 3: Substitute into the Equation
Substituting these into the original equation gives:
\[ (\cos \beta \cos \gamma + \sin \beta \sin \gamma) + (\cos \gamma \cos \alpha + \sin \gamma \sin \alpha) + (\cos \alpha \cos \beta + \sin \alpha \sin \beta) = -\frac{3}{2} \]

### Step 4: Combine Like Terms
Combining the terms, we have:
\[ \cos \beta \cos \gamma + \cos \gamma \cos \alpha + \cos \alpha \cos \beta + \sin \beta \sin \gamma + \sin \gamma \sin \alpha + \sin \alpha \sin \beta = -\frac{3}{2} \]

### Step 5: Use the Identity for the Sum of Cosines
We can rewrite the left-hand side using the identity for the sum of cosines:
\[ \cos \beta \cos \gamma + \cos \gamma \cos \alpha + \cos \alpha \cos \beta = \frac{1}{2} \left( (\cos \alpha + \cos \beta + \cos \gamma)^2 - (\cos^2 \alpha + \cos^2 \beta + \cos^2 \gamma) \right) \]
And similarly for the sine terms:
\[ \sin \beta \sin \gamma + \sin \gamma \sin \alpha + \sin \alpha \sin \beta = \frac{1}{2} \left( (\sin \alpha + \sin \beta + \sin \gamma)^2 - (\sin^2 \alpha + \sin^2 \beta + \sin^2 \gamma) \right) \]

### Step 6: Analyze the Result
From the given equation, we know that:
\[ \cos \alpha + \cos \beta + \cos \gamma = 0 \quad \text{and} \quad \sin \alpha + \sin \beta + \sin \gamma = 0 \]
Thus, both statements A and B are satisfied.

### Conclusion
Both statements A and B are true based on the given conditions.