If `cos (beta-gamma)+cos(gamma-alpha)+cos(alpha-beta)=-(3)/(2)` then

Options

A. `Sigma cos alpha=0`
B. `Sigma sin alpha=0`
C. `Sigma cos alpha sin alpha=0`
D. `Sigma(cos alpha + sin alpha)=0`

Answer

To solve the equation \( \cos(\beta - \gamma) + \cos(\gamma - \alpha) + \cos(\alpha - \beta) = -\frac{3}{2} \), we will use the cosine difference identity and manipulate the equation step by step.

### Step 1: Apply the Cosine Difference Identity
We know that:
\[ \cos(A - B) = \cos A \cos B + \sin A \sin B \]
Using this identity, we can rewrite each cosine term:
\[ \cos(\beta - \gamma) = \cos \beta \cos \gamma + \sin \beta \sin \gamma \]
\[ \cos(\gamma - \alpha) = \cos \gamma \cos \alpha + \sin \gamma \sin \alpha \]
\[ \cos(\alpha - \beta) = \cos \alpha \cos \beta + \sin \alpha \sin \beta \]

### Step 2: Substitute into the Equation
Substituting these identities 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 3: Combine Like Terms
Now, we can combine all the terms:
\[ \cos \beta \cos \gamma + \cos \gamma \cos \alpha + \cos \alpha \cos \beta + \sin \beta \sin \gamma + \sin \gamma \sin \alpha + \sin \alpha \sin \beta = -\frac{3}{2} \]

### Step 4: Rearranging the Equation
We can rearrange the equation to group cosine and sine terms:
\[ \cos \beta \cos \gamma + \cos \gamma \cos \alpha + \cos \alpha \cos \beta + \sin \beta \sin \gamma + \sin \gamma \sin \alpha + \sin \alpha \sin \beta + \frac{3}{2} = 0 \]

### Step 5: Recognizing the Structure
Notice that the left-hand side can be recognized as a sum of squares:
\[ \left(\sin \alpha + \sin \beta + \sin \gamma\right)^2 + \left(\cos \alpha + \cos \beta + \cos \gamma\right)^2 = 0 \]

### Step 6: Conclude the Values
Since both terms are squares, they can only equal zero if:
\[ \sin \alpha + \sin \beta + \sin \gamma = 0 \]
\[ \cos \alpha + \cos \beta + \cos \gamma = 0 \]

Thus, we conclude that:
- The sum of the sine of the angles is zero.
- The sum of the cosine of the angles is zero.

### Final Answer
The values of \( \alpha, \beta, \gamma \) must satisfy:
\[ \sin \alpha + \sin \beta + \sin \gamma = 0 \quad \text{and} \quad \cos \alpha + \cos \beta + \cos \gamma = 0 \]