Let `cos(alpha-beta) + cos(beta - gamma) + cos(gamma - alpha)= -3/2`, then the value of `cos alpha + cos beta + cos gamma` is

Answer

To solve the equation \( \cos(\alpha - \beta) + \cos(\beta - \gamma) + \cos(\gamma - \alpha) = -\frac{3}{2} \), we will proceed step by step.

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

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

### Step 3: Rearranging
Rearranging the equation, we can group the cosine and sine terms:
\[ \cos \alpha \cos \beta + \cos \beta \cos \gamma + \cos \gamma \cos \alpha + \sin \alpha \sin \beta + \sin \beta \sin \gamma + \sin \gamma \sin \alpha = -\frac{3}{2} \]

### Step 4: Recognizing the Maximum Value of Cosine
The maximum value of \( \cos(x) \) is 1. Therefore, the minimum value of \( \cos(\alpha - \beta) + \cos(\beta - \gamma) + \cos(\gamma - \alpha) \) occurs when each cosine term is at its minimum, which is -1.

### Step 5: Analyzing the Given Condition
The equation \( \cos(\alpha - \beta) + \cos(\beta - \gamma) + \cos(\gamma - \alpha) = -\frac{3}{2} \) indicates that the angles are such that the sum of the cosines is less than the minimum value of -3. This suggests that the angles are positioned in such a way that they are not aligned.

### Step 6: Finding the Value of \( \cos \alpha + \cos \beta + \cos \gamma \)
From the properties of trigonometric functions and the symmetry in the angles, we can conclude that:
\[ \cos \alpha + \cos \beta + \cos \gamma = 0 \]

Thus, the final answer is:
\[ \cos \alpha + \cos \beta + \cos \gamma = 0 \]