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

Options

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

Answer

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

### Step-by-Step Solution:

1. **Understanding Direction Cosines**:
The direction cosines of a line making angles \( \frac{\alpha}{2}, \frac{\beta}{2}, \frac{\gamma}{2} \) with the x, y, and z axes are given by:
\[ l = \cos\left(\frac{\alpha}{2}\right), \quad m = \cos\left(\frac{\beta}{2}\right), \quad n = \cos\left(\frac{\gamma}{2}\right) \]

2. **Using the Property of Direction Cosines**:
The sum of the squares of the direction cosines is equal to 1:
\[ \cos^2\left(\frac{\alpha}{2}\right) + \cos^2\left(\frac{\beta}{2}\right) + \cos^2\left(\frac{\gamma}{2}\right) = 1 \]

3. **Using the Cosine Double Angle Formula**:
We know the relation:
\[ \cos \theta = 2 \cos^2\left(\frac{\theta}{2}\right) - 1 \]
Therefore, we can express \( \cos \alpha, \cos \beta, \cos \gamma \) in terms of \( \cos\left(\frac{\alpha}{2}\right), \cos\left(\frac{\beta}{2}\right), \cos\left(\frac{\gamma}{2}\right) \):
\[ \cos \alpha = 2 \cos^2\left(\frac{\alpha}{2}\right) - 1 \]
\[ \cos \beta = 2 \cos^2\left(\frac{\beta}{2}\right) - 1 \]
\[ \cos \gamma = 2 \cos^2\left(\frac{\gamma}{2}\right) - 1 \]

4. **Substituting Back**:
Now, substituting these into the expression \( \cos \alpha + \cos \beta + \cos \gamma \):
\[ \cos \alpha + \cos \beta + \cos \gamma = \left(2 \cos^2\left(\frac{\alpha}{2}\right) - 1\right) + \left(2 \cos^2\left(\frac{\beta}{2}\right) - 1\right) + \left(2 \cos^2\left(\frac{\gamma}{2}\right) - 1\right) \]
Simplifying this gives:
\[ = 2\left(\cos^2\left(\frac{\alpha}{2}\right) + \cos^2\left(\frac{\beta}{2}\right) + \cos^2\left(\frac{\gamma}{2}\right)\right) - 3 \]

5. **Using the Property of Direction Cosines Again**:
Since we know that \( \cos^2\left(\frac{\alpha}{2}\right) + \cos^2\left(\frac{\beta}{2}\right) + \cos^2\left(\frac{\gamma}{2}\right) = 1 \):
\[ \cos \alpha + \cos \beta + \cos \gamma = 2(1) - 3 = 2 - 3 = -1 \]

### Final Answer:
Thus, we conclude that:
\[ \cos \alpha + \cos \beta + \cos \gamma = -1 \]