A line makes angles `alpha, beta, gamma` with X, Y, Z axes respectively. If `alpha=beta` and `gamma=45^(@)`, then `alpha=`

Options

A. `0^(@)`
B. `30^(@)`
C. `60^(@)`
D. `90^(@)`

Answer

To solve the problem, we need to find the value of the angle \( \alpha \) given the conditions that \( \alpha = \beta \) and \( \gamma = 45^\circ \).

### Step-by-Step Solution:

1. **Understanding the Angles**:
The angles \( \alpha, \beta, \gamma \) are the angles that a line makes with the X, Y, and Z axes respectively. We know that \( \alpha = \beta \) and \( \gamma = 45^\circ \).

2. **Using the Cosine Relation**:
The relationship between these angles can be expressed using the equation:
\[ \cos^2 \alpha + \cos^2 \beta + \cos^2 \gamma = 1 \]
Since \( \alpha = \beta \), we can rewrite the equation as:
\[ 2\cos^2 \alpha + \cos^2 \gamma = 1 \]

3. **Substituting the Value of \( \gamma \)**:
We know that \( \gamma = 45^\circ \). Therefore, we can find \( \cos \gamma \):
\[ \cos 45^\circ = \frac{1}{\sqrt{2}} \]
Thus, \( \cos^2 \gamma = \left(\frac{1}{\sqrt{2}}\right)^2 = \frac{1}{2} \).

4. **Substituting into the Equation**:
Now we substitute \( \cos^2 \gamma \) into our equation:
\[ 2\cos^2 \alpha + \frac{1}{2} = 1 \]

5. **Solving for \( \cos^2 \alpha \)**:
Rearranging the equation gives:
\[ 2\cos^2 \alpha = 1 - \frac{1}{2} = \frac{1}{2} \]
Dividing both sides by 2:
\[ \cos^2 \alpha = \frac{1}{4} \]

6. **Finding \( \cos \alpha \)**:
Taking the square root of both sides, we get:
\[ \cos \alpha = \pm \frac{1}{2} \]

7. **Finding the Angles**:
The angles corresponding to \( \cos \alpha = \frac{1}{2} \) are:
\[ \alpha = 60^\circ \quad \text{(1st quadrant)} \]
The angle corresponding to \( \cos \alpha = -\frac{1}{2} \) is:
\[ \alpha = 120^\circ \quad \text{(2nd quadrant)} \]

### Conclusion:
Thus, the possible values of \( \alpha \) are \( 60^\circ \) and \( 120^\circ \).