If a line makes angles `alpha,beta,gamma` with co-ordinate axes, then `cos2alpha+cos 2 beta+cos2gamma=`

Options

A. 0
B. -1
C. 2
D. 1

Answer

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

### Step-by-Step Solution:

1. **Understanding Direction Cosines**:
The direction cosines of a line making angles \( \alpha, \beta, \gamma \) with the x, y, and z axes respectively are given by:
\[ \cos \alpha, \cos \beta, \cos \gamma \]

2. **Using the Property of Direction Cosines**:
There is a fundamental property of direction cosines:
\[ \cos^2 \alpha + \cos^2 \beta + \cos^2 \gamma = 1 \]

3. **Relating Cosine of Double Angles**:
We can express \( \cos 2\theta \) in terms of \( \cos^2 \theta \):
\[ \cos 2\theta = 2\cos^2 \theta - 1 \]
Therefore, we can write:
\[ \cos 2\alpha = 2\cos^2 \alpha - 1 \]
\[ \cos 2\beta = 2\cos^2 \beta - 1 \]
\[ \cos 2\gamma = 2\cos^2 \gamma - 1 \]

4. **Summing the Cosine of Double Angles**:
Now, we can sum 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:
\[ = 2(\cos^2 \alpha + \cos^2 \beta + \cos^2 \gamma) - 3 \]

5. **Substituting the Property**:
Using the property from step 2, we substitute \( \cos^2 \alpha + \cos^2 \beta + \cos^2 \gamma = 1 \):
\[ = 2(1) - 3 \]
\[ = 2 - 3 = -1 \]

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

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