The difference of tangents of the angles which the lines given by
`x^(2)(tan^(2)alpha+cos^(2) alpha)-2xytanalpha+y^(2)sin^(2)alpha=0` make with the x -axis is

Options

A. 1
B. 2
C. `sin2 theta`
D. `2 sin theta`

Answer

To find the difference of the tangents of the angles which the lines given by the equation

\[ x^2 (\tan^2 \alpha + \cos^2 \alpha) - 2xy \tan \alpha + y^2 \sin^2 \alpha = 0 \]

make with the x-axis, we can follow these steps:

### Step 1: Rewrite the Equation
We start with the given equation and divide through by \( y \) to express it in terms of \( m = \frac{y}{x} \):

\[ x^2 (\tan^2 \alpha + \cos^2 \alpha) - 2x \cdot \frac{y}{x} \tan \alpha + \left(\frac{y}{x}\right)^2 \sin^2 \alpha \cdot x^2 = 0 \]

This simplifies to:

\[ x^2 \left( \tan^2 \alpha + \cos^2 \alpha \right) - 2x^2 m \tan \alpha + x^2 m^2 \sin^2 \alpha = 0 \]

### Step 2: Form a Quadratic Equation
We can factor out \( x^2 \):

\[ x^2 \left( \tan^2 \alpha + \cos^2 \alpha - 2m \tan \alpha + m^2 \sin^2 \alpha \right) = 0 \]

This gives us a quadratic equation in \( m \):

\[ m^2 \sin^2 \alpha - 2m \tan \alpha + (\tan^2 \alpha + \cos^2 \alpha) = 0 \]

### Step 3: Identify Coefficients
From the quadratic equation \( am^2 + bm + c = 0 \), we identify:

- \( a = \sin^2 \alpha \)
- \( b = -2 \tan \alpha \)
- \( c = \tan^2 \alpha + \cos^2 \alpha \)

### Step 4: Calculate Roots
Using the quadratic formula, the roots \( m_1 \) and \( m_2 \) are given by:

\[ m = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]

Substituting our coefficients:

\[ m = \frac{2 \tan \alpha \pm \sqrt{(-2 \tan \alpha)^2 - 4 \sin^2 \alpha (\tan^2 \alpha + \cos^2 \alpha)}}{2 \sin^2 \alpha} \]

### Step 5: Find the Difference of Roots
The difference of the roots \( m_1 - m_2 \) can be calculated as:

\[ m_1 - m_2 = \sqrt{(m_1 + m_2)^2 - 4m_1 m_2} \]

Using Vieta's formulas:

- \( m_1 + m_2 = \frac{2 \tan \alpha}{\sin^2 \alpha} \)
- \( m_1 m_2 = \frac{\tan^2 \alpha + \cos^2 \alpha}{\sin^2 \alpha} \)

### Step 6: Substitute and Simplify
Substituting these values back into the difference formula, we can simplify it to find \( m_1 - m_2 \).

After simplification, we find that:

\[ m_1 - m_2 = 2 \]

### Conclusion
Thus, the difference of the tangents of the angles which the lines make with the x-axis is:

\[ \boxed{2} \]