If `cos^(-1) alpha + cos^(-1) beta + cos^(-1) gamma = 3pi`, then `alpha (beta + gamma) + beta(gamma + alpha) + gamma(alpha + beta)` equal to

Options

A. `0`
B. `1`
C. `6`
D. `12`

Explanation

To solve the problem, we start with the given equation:

\[ \cos^{-1} \alpha + \cos^{-1} \beta + \cos^{-1} \gamma = 3\pi \]

### Step 1: Analyze the Range of Inverse Cosine
The range of \(\cos^{-1} x\) is from \(0\) to \(\pi\). Therefore, for the sum of three inverse cosine functions to equal \(3\pi\), each individual term must equal \(\pi\).

### Step 2: Set Each Inverse Cosine to \(\pi\)
From the analysis, we conclude that:
\[ \cos^{-1} \alpha = \pi, \quad \cos^{-1} \beta = \pi, \quad \cos^{-1} \gamma = \pi \]

### Step 3: Find the Values of \(\alpha\), \(\beta\), and \(\gamma\)
The equation \(\cos^{-1} x = \pi\) implies:
\[ \alpha = \cos(\pi) = -1, \quad \beta = \cos(\pi) = -1, \quad \gamma = \cos(\pi) = -1 \]

### Step 4: Substitute Values into the Expression
Now we need to evaluate the expression:
\[ \alpha(\beta + \gamma) + \beta(\gamma + \alpha) + \gamma(\alpha + \beta) \]

Substituting \(\alpha = -1\), \(\beta = -1\), and \(\gamma = -1\):
\[ = (-1)(-1 + -1) + (-1)(-1 + -1) + (-1)(-1 + -1) \]

### Step 5: Simplify Each Term
Calculating each term:
\[ = (-1)(-2) + (-1)(-2) + (-1)(-2) \]
\[ = 2 + 2 + 2 \]

### Step 6: Final Calculation
Adding these values together:
\[ = 6 \]

### Conclusion
Thus, the value of \(\alpha(\beta + \gamma) + \beta(\gamma + \alpha) + \gamma(\alpha + \beta)\) is:
\[ \boxed{6} \]