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 coordinate axes, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding Direction Cosines**:
The direction cosines of a line are denoted as \( l \), \( m \), and \( n \), which correspond to the angles \( \alpha \), \( \beta \), and \( \gamma \) that the line makes with the x-axis, y-axis, and z-axis respectively.

2. **Equally Inclined Condition**:
Since the line is equally inclined to the axes, we have:
\[ \cos \alpha = \cos \beta = \cos \gamma \]
This implies:
\[ l = m = n \]

3. **Using the Identity**:
We know from the properties of direction cosines that:
\[ l^2 + m^2 + n^2 = 1 \]
Substituting \( l = m = n \) into this equation gives:
\[ l^2 + l^2 + l^2 = 1 \]
Simplifying this, we get:
\[ 3l^2 = 1 \]

4. **Solving for \( l \)**:
Dividing both sides by 3:
\[ l^2 = \frac{1}{3} \]
Taking the square root of both sides:
\[ l = \pm \frac{1}{\sqrt{3}} \]

5. **Finding Direction Cosines**:
Since \( l = m = n \), we have:
\[ l = m = n = \frac{1}{\sqrt{3}} \quad \text{or} \quad l = m = n = -\frac{1}{\sqrt{3}} \]
Therefore, the direction cosines of the line are:
\[ \left( \frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}} \right) \quad \text{and} \quad \left( -\frac{1}{\sqrt{3}}, -\frac{1}{\sqrt{3}}, -\frac{1}{\sqrt{3}} \right) \]

### Final Answer:
The direction cosines of a line equally inclined to the axes are:
\[ \left( \frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}} \right) \quad \text{and} \quad \left( -\frac{1}{\sqrt{3}}, -\frac{1}{\sqrt{3}}, -\frac{1}{\sqrt{3}} \right) \]