The difference of the tangents of the angles which the lines `x^(2)(sec^(2)-sin^(2)theta)-2xy tan theta +y^(2)sin^(2) theta=0` make with X-axis, is

Options

A. `2 tan theta`
B. 2
C. `2 cot theta`
D. `sin 2 theta`

Answer

To solve the problem of finding the difference of the tangents of the angles which the lines represented by the equation

\[ x^2 \sec^2 \theta - \sin^2 \theta - 2xy \tan \theta + y^2 \sin^2 \theta = 0 \]

make with the X-axis, we will follow these steps:

### Step 1: Identify the coefficients
We start by rewriting the given equation in the standard form of a pair of straight lines:

\[ Ax^2 + 2Hxy + By^2 = 0 \]

From the given equation, we can identify:
- \( A = \sec^2 \theta \)
- \( B = \sin^2 \theta \)
- \( 2H = -2 \tan \theta \) which gives \( H = -\tan \theta \)

### Step 2: Use the formulas for slopes
For the pair of lines, the slopes \( m_1 \) and \( m_2 \) can be calculated using the formulas:
- \( m_1 + m_2 = -\frac{2H}{B} \)
- \( m_1 m_2 = \frac{A}{B} \)

Substituting the values we identified:
1. \( m_1 + m_2 = -\frac{2(-\tan \theta)}{\sin^2 \theta} = \frac{2\tan \theta}{\sin^2 \theta} = 2 \frac{\tan \theta}{\sin^2 \theta} \)
2. \( m_1 m_2 = \frac{\sec^2 \theta}{\sin^2 \theta} \)

### Step 3: Find the difference of the slopes
To find the difference of the slopes \( m_1 - m_2 \), we use the identity:
\[ (m_1 - m_2)^2 = (m_1 + m_2)^2 - 4m_1 m_2 \]

Substituting the values:
\[ (m_1 - m_2)^2 = \left(2 \frac{\tan \theta}{\sin^2 \theta}\right)^2 - 4 \left(\frac{\sec^2 \theta}{\sin^2 \theta}\right) \]

### Step 4: Simplify the expression
Calculating \( (m_1 - m_2)^2 \):
1. \( (m_1 + m_2)^2 = 4 \frac{\tan^2 \theta}{\sin^4 \theta} \)
2. \( 4m_1 m_2 = 4 \cdot \frac{\sec^2 \theta}{\sin^2 \theta} = 4 \cdot \frac{1}{\cos^2 \theta \sin^2 \theta} \)

Thus,
\[ (m_1 - m_2)^2 = 4 \frac{\tan^2 \theta}{\sin^4 \theta} - 4 \cdot \frac{1}{\cos^2 \theta \sin^2 \theta} \]

### Step 5: Factor out the common terms
Factoring out \( 4 \):
\[ (m_1 - m_2)^2 = 4 \left( \frac{\tan^2 \theta}{\sin^4 \theta} - \frac{1}{\cos^2 \theta \sin^2 \theta} \right) \]

### Step 6: Find the difference of the tangents
To find \( m_1 - m_2 \):
\[ m_1 - m_2 = \sqrt{4} = 2 \]

Thus, the difference of the tangents of the angles which the lines make with the X-axis is \( 2 \).

### Final Answer
The difference of the tangents of the angles which the lines make with the X-axis is \( \boxed{2} \).