The expression `cos^(2)(alpha + beta + gamma) + cos^(2)(beta + gamma) + cos^(2)alpha - 2cos alpha cos(beta + gamma)cos(alpha + beta + gamma)` is (a) independent of `alpha` (b) independent of `beta` (c) dependent on `gamma` only (d) dependent on `alpha, beta, gamma`

Options

A. independent of `alpha`
B. independent of `beta`
C. dependent on `gamma` only
D. dependent on `alpha, beta, gamma`

Explanation

To solve the expression

\[ \cos^2(\alpha + \beta + \gamma) + \cos^2(\beta + \gamma) + \cos^2 \alpha - 2 \cos \alpha \cos(\beta + \gamma) \cos(\alpha + \beta + \gamma), \]

we will simplify it step by step.

### Step 1: Rewrite the expression

The expression can be rewritten as:

\[ E = \cos^2(\alpha + \beta + \gamma) + \cos^2(\beta + \gamma) + \cos^2 \alpha - 2 \cos \alpha \cos(\beta + \gamma) \cos(\alpha + \beta + \gamma). \]

### Step 2: Apply the identity for \(2 \cos A \cos B\)

We know that:

\[ 2 \cos A \cos B = \cos(A + B) + \cos(A - B). \]

In our case, let \(A = \alpha\) and \(B = \beta + \gamma\). Thus, we can rewrite the term:

\[ -2 \cos \alpha \cos(\beta + \gamma) = -\left(\cos(\alpha + \beta + \gamma) + \cos(\alpha - (\beta + \gamma))\right). \]

### Step 3: Substitute back into the expression

Substituting this back into the expression, we have:

\[ E = \cos^2(\alpha + \beta + \gamma) + \cos^2(\beta + \gamma) + \cos^2 \alpha - \left(\cos(\alpha + \beta + \gamma) + \cos(\alpha - (\beta + \gamma))\right). \]

### Step 4: Combine like terms

Now, we can combine the terms:

\[ E = \cos^2(\alpha + \beta + \gamma) - \cos(\alpha + \beta + \gamma) + \cos^2(\beta + \gamma) + \cos^2 \alpha - \cos(\alpha - (\beta + \gamma)). \]

### Step 5: Analyze the expression

Notice that the expression consists of terms involving \(\cos^2\) and \(\cos\) of combinations of \(\alpha\), \(\beta\), and \(\gamma\).

### Step 6: Evaluate dependencies

To check the dependencies of the expression on \(\alpha\), \(\beta\), and \(\gamma\):

1. **Independent of \(\alpha\)**: The expression does not contain any linear terms of \(\alpha\) that would affect the value of \(E\).
2. **Independent of \(\beta\)**: Similarly, there are no linear terms of \(\beta\) that affect the value of \(E\).
3. **Dependent on \(\gamma\)**: The expression does depend on \(\gamma\) through the cosine terms.

### Conclusion

Thus, the expression is independent of both \(\alpha\) and \(\beta\), and it is dependent on \(\gamma\) only.

### Final Answer

The correct options are:
- (a) independent of \(\alpha\)
- (b) independent of \(\beta\)
- (c) dependent on \(\gamma\) only