If a line makes `30^@` with the positive direction of X-axis. `anglebeta` with the positive direction of Y-axis and `anglegamma` with the positive direction of Z-axis, then what is `cos^2beta+cos^2gamma` equal to?

Options

A. `1//4`
B. `1//2`
C. `3//4`
D. `1`

Answer

To solve the problem, we need to find the value of \( \cos^2 \beta + \cos^2 \gamma \) given that a line makes a \( 30^\circ \) angle with the positive direction of the X-axis, an angle \( \beta \) with the positive direction of the Y-axis, and an angle \( \gamma \) with the positive direction of the Z-axis.

### Step-by-Step Solution:

1. **Understanding Direction Cosines**:
- The direction cosines of a line making angles \( \alpha \), \( \beta \), and \( \gamma \) with the X, Y, and Z axes respectively are given by:
\[ \cos \alpha, \cos \beta, \cos \gamma \]
- In this case, \( \alpha = 30^\circ \).

2. **Calculate \( \cos \alpha \)**:
- We know:
\[ \cos 30^\circ = \frac{\sqrt{3}}{2} \]

3. **Using the Property of Direction Cosines**:
- The sum of the squares of the direction cosines is equal to 1:
\[ \cos^2 \alpha + \cos^2 \beta + \cos^2 \gamma = 1 \]

4. **Substituting \( \cos^2 \alpha \)**:
- Substitute \( \cos^2 30^\circ \) into the equation:
\[ \left(\frac{\sqrt{3}}{2}\right)^2 + \cos^2 \beta + \cos^2 \gamma = 1 \]
- This simplifies to:
\[ \frac{3}{4} + \cos^2 \beta + \cos^2 \gamma = 1 \]

5. **Rearranging the Equation**:
- Rearranging gives us:
\[ \cos^2 \beta + \cos^2 \gamma = 1 - \frac{3}{4} \]
- This simplifies to:
\[ \cos^2 \beta + \cos^2 \gamma = \frac{1}{4} \]

6. **Final Result**:
- Therefore, the value of \( \cos^2 \beta + \cos^2 \gamma \) is:
\[ \cos^2 \beta + \cos^2 \gamma = \frac{1}{4} \]

### Conclusion:
The final answer is:
\[ \cos^2 \beta + \cos^2 \gamma = \frac{1}{4} \]