The direction cosines of the lines bisecting the angle between the line whose direction cosines are `l_1, m_1, n_1 and l_2, m_2, n_2` and the angle between these lines is `theta`, are

Options

A. `(l_1+l_2)/(2sin((theta)/(2))), (m_1+m_2)/(2sin((theta)/(2))), (n_1+n_2)/(2sin((theta)/(2)))`
B. `(l_1+l_2)/(2cos((theta)/(2))), (m_1+m_2)/(2cos((theta)/(2))), (n_1+n_2)/(2cos((theta)/(2)))`
C. `(l_1-l_2)/(2sin((theta)/(2))), (m_1-m_2)/(2sin((theta)/(2))), (n_1-n_2)/(2sin((theta)/(2)))`
D. `(l_1-l_2)/(2cos((theta)/(2))), (m_1-m_2)/(2cos((theta)/(2))), (n_1-n_2)/(2cos((theta)/(2)))`

Answer

To find the direction cosines of the line bisecting the angle between two lines with direction cosines \( l_1, m_1, n_1 \) and \( l_2, m_2, n_2 \), we can follow these steps:

### Step 1: Understand the Direction Cosines
The direction cosines of a line are defined as the cosines of the angles that the line makes with the coordinate axes. For the lines represented by direction cosines \( (l_1, m_1, n_1) \) and \( (l_2, m_2, n_2) \), we know that:
\[ l_1^2 + m_1^2 + n_1^2 = 1 \]
\[ l_2^2 + m_2^2 + n_2^2 = 1 \]

### Step 2: Use the Cosine of the Angle Formula
The cosine of the angle \( \theta \) between the two lines can be calculated using the formula:
\[ \cos \theta = l_1 l_2 + m_1 m_2 + n_1 n_2 \]

### Step 3: Find the Direction Ratios for the Bisector
The direction ratios for the line bisecting the angle between the two lines can be given as:
\[ (l_1 + l_2, m_1 + m_2, n_1 + n_2) \]

### Step 4: Calculate the Magnitude of the Direction Ratios
The magnitude of the direction ratios is calculated as:
\[ \text{Magnitude} = \sqrt{(l_1 + l_2)^2 + (m_1 + m_2)^2 + (n_1 + n_2)^2} \]
Expanding this, we get:
\[ = \sqrt{(l_1^2 + l_2^2 + 2l_1l_2) + (m_1^2 + m_2^2 + 2m_1m_2) + (n_1^2 + n_2^2 + 2n_1n_2)} \]
Using the fact that \( l_1^2 + m_1^2 + n_1^2 = 1 \) and \( l_2^2 + m_2^2 + n_2^2 = 1 \):
\[ = \sqrt{1 + 1 + 2(l_1l_2 + m_1m_2 + n_1n_2)} = \sqrt{2 + 2\cos \theta} \]

### Step 5: Calculate the Direction Cosines
The direction cosines of the bisector can now be calculated as:
\[ \text{Direction cosines} = \left( \frac{l_1 + l_2}{\sqrt{2 + 2\cos \theta}}, \frac{m_1 + m_2}{\sqrt{2 + 2\cos \theta}}, \frac{n_1 + n_2}{\sqrt{2 + 2\cos \theta}} \right) \]

### Step 6: Consider the Other Bisector
For the other bisector (the one that bisects the angle in the opposite direction), the direction ratios would be:
\[ (l_1 - l_2, m_1 - m_2, n_1 - n_2) \]
And the magnitude would similarly be:
\[ \sqrt{(l_1 - l_2)^2 + (m_1 - m_2)^2 + (n_1 - n_2)^2} \]
This leads to direction cosines:
\[ \text{Direction cosines} = \left( \frac{l_1 - l_2}{\sqrt{2 - 2\cos \theta}}, \frac{m_1 - m_2}{\sqrt{2 - 2\cos \theta}}, \frac{n_1 - n_2}{\sqrt{2 - 2\cos \theta}} \right) \]

### Summary of Results
Thus, the direction cosines of the line bisecting the angle between the two lines are:
1. For the first bisector:
\[ \left( \frac{l_1 + l_2}{\sqrt{2 + 2\cos \theta}}, \frac{m_1 + m_2}{\sqrt{2 + 2\cos \theta}}, \frac{n_1 + n_2}{\sqrt{2 + 2\cos \theta}} \right) \]
2. For the second bisector:
\[ \left( \frac{l_1 - l_2}{\sqrt{2 - 2\cos \theta}}, \frac{m_1 - m_2}{\sqrt{2 - 2\cos \theta}}, \frac{n_1 - n_2}{\sqrt{2 - 2\cos \theta}} \right) \]