If a line makes the angles `alpha,beta,gamma` with the axes, then what is the value of `1+cos2alpha+cos2beta+cosgamma` equal to

Options

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

Explanation

To solve the problem, we need to find the value of the expression \(1 + \cos 2\alpha + \cos 2\beta + \cos 2\gamma\) given that a line makes angles \(\alpha\), \(\beta\), and \(\gamma\) with the x, y, and z axes respectively.

### Step-by-step Solution:

1. **Understanding Direction Cosines**:
The direction cosines of a line making angles \(\alpha\), \(\beta\), and \(\gamma\) with the axes are given by:
\[ \cos \alpha, \cos \beta, \cos \gamma \]
According to the property of direction cosines, we have:
\[ \cos^2 \alpha + \cos^2 \beta + \cos^2 \gamma = 1 \]

2. **Using the Cosine Double Angle Formula**:
We can express \(\cos 2\theta\) using the double angle formula:
\[ \cos 2\theta = 2\cos^2 \theta - 1 \]
Applying this to our angles, 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. **Substituting into the Expression**:
Now, substituting these into the expression \(1 + \cos 2\alpha + \cos 2\beta + \cos 2\gamma\):
\[ 1 + (2\cos^2 \alpha - 1) + (2\cos^2 \beta - 1) + (2\cos^2 \gamma - 1) \]
Simplifying this gives:
\[ 1 + 2\cos^2 \alpha + 2\cos^2 \beta + 2\cos^2 \gamma - 3 \]
\[ = 2\cos^2 \alpha + 2\cos^2 \beta + 2\cos^2 \gamma - 2 \]
\[ = 2(\cos^2 \alpha + \cos^2 \beta + \cos^2 \gamma - 1) \]

4. **Using the Property of Direction Cosines**:
From the property of direction cosines, we know that:
\[ \cos^2 \alpha + \cos^2 \beta + \cos^2 \gamma = 1 \]
Therefore, substituting this into our expression:
\[ = 2(1 - 1) = 2 \times 0 = 0 \]

5. **Final Result**:
Thus, the value of \(1 + \cos 2\alpha + \cos 2\beta + \cos 2\gamma\) is:
\[ \boxed{0} \]