The direction cosines of a line equally inclined to the axes are:

Options

A. `(1)/(3),(1)/(3) , (1)/(3)`
B. `-(1)/(3), (1)/(3) , (1)/(3)`
C. `(1)/(sqrt3),(1)/(sqrt3), (1)/(sqrt3)`
D. `-(1)/(sqrt3),- (1)/(sqrt3), -(1)/(sqrt3)`

Answer

To find the direction cosines of a line that is equally inclined to the axes, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding Direction Cosines**:
The direction cosines of a line are the cosines of the angles that the line makes with the coordinate axes. If a line is equally inclined to the x, y, and z axes, then the angles made with each axis are equal.

2. **Setting Up the Angles**:
Let the angles made by the line with the x, y, and z axes be denoted as \( \alpha \), \( \beta \), and \( \gamma \). Since the line is equally inclined to the axes, we have:
\[ \alpha = \beta = \gamma \]

3. **Direction Cosines Representation**:
The direction cosines \( l \), \( m \), and \( n \) corresponding to the angles \( \alpha \), \( \beta \), and \( \gamma \) can be expressed as:
\[ l = \cos(\alpha), \quad m = \cos(\beta), \quad n = \cos(\gamma) \]
Since \( \alpha = \beta = \gamma \), we can denote:
\[ l = m = n = k \]

4. **Using the Identity for Direction Cosines**:
The direction cosines satisfy the identity:
\[ l^2 + m^2 + n^2 = 1 \]
Substituting \( l = m = n = k \) into the identity gives:
\[ k^2 + k^2 + k^2 = 1 \]
This simplifies to:
\[ 3k^2 = 1 \]

5. **Solving for \( k \)**:
Dividing both sides by 3, we find:
\[ k^2 = \frac{1}{3} \]
Taking the square root of both sides results in:
\[ k = \pm \frac{1}{\sqrt{3}} \]

6. **Finding Direction Cosines**:
Since \( l = m = n = k \), the direction cosines of the line can be expressed as:
\[ (l, m, n) = \left( \frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}} \right) \quad \text{and} \quad (l, m, n) = \left( -\frac{1}{\sqrt{3}}, -\frac{1}{\sqrt{3}}, -\frac{1}{\sqrt{3}} \right) \]

7. **Conclusion**:
The direction cosines of a line equally inclined to the axes are:
\[ \left( \frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}} \right) \quad \text{and} \quad \left( -\frac{1}{\sqrt{3}}, -\frac{1}{\sqrt{3}}, -\frac{1}{\sqrt{3}} \right) \]
Therefore, the correct option is:
\[ \frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}} \]