If the direction angles of a line are `alpha, beta and gamma` respectively, then `cos 2 alpha + cos 2 beta + cos 2gamma` is equal to

Options

A. `1`
B. `-1`
C. `2`
D. `-2`

Answer

To solve the problem, we need to find the value of \( \cos 2\alpha + \cos 2\beta + \cos 2\gamma \) given that \( \alpha, \beta, \gamma \) are the direction angles of a line.

### Step-by-Step Solution:

1. **Recall the identity for cosine of double angles**:
\[ \cos 2\theta = 2\cos^2 \theta - 1 \]
We will use this identity for \( \alpha, \beta, \) and \( \gamma \).

2. **Apply the identity**:
Using the identity for each angle, we have:
\[ \cos 2\alpha = 2\cos^2 \alpha - 1 \]
\[ \cos 2\beta = 2\cos^2 \beta - 1 \]
\[ \cos 2\gamma = 2\cos^2 \gamma - 1 \]

3. **Combine the expressions**:
Now, we can combine these expressions:
\[ \cos 2\alpha + \cos 2\beta + \cos 2\gamma = (2\cos^2 \alpha - 1) + (2\cos^2 \beta - 1) + (2\cos^2 \gamma - 1) \]
Simplifying this gives:
\[ = 2\cos^2 \alpha + 2\cos^2 \beta + 2\cos^2 \gamma - 3 \]

4. **Use the property of direction cosines**:
We know that for direction angles \( \alpha, \beta, \gamma \):
\[ \cos^2 \alpha + \cos^2 \beta + \cos^2 \gamma = 1 \]

5. **Substitute this property into the equation**:
Substituting this into our combined expression:
\[ = 2(1) - 3 = 2 - 3 = -1 \]

6. **Final Result**:
Therefore, we conclude that:
\[ \cos 2\alpha + \cos 2\beta + \cos 2\gamma = -1 \]

### Conclusion:
The final answer is:
\[ \cos 2\alpha + \cos 2\beta + \cos 2\gamma = -1 \]