Show that the tangent of an angle between the lines `(x)/(a)+(y)/(b)=1` and `(x)/(a)-(y)/(b)=1` and `(2ab)/(a^(2)-b^(2))`.

Explanation

To show that the tangent of the angle between the lines \(\frac{x}{a} + \frac{y}{b} = 1\) and \(\frac{x}{a} - \frac{y}{b} = 1\) is \(\frac{2ab}{a^2 - b^2}\), we will follow these steps:

### Step 1: Convert the equations of the lines to slope-intercept form
The given equations are:
1. \(\frac{x}{a} + \frac{y}{b} = 1\)
2. \(\frac{x}{a} - \frac{y}{b} = 1\)

We can rewrite these equations in the form \(y = mx + c\) to find their slopes.

**For the first line:**
\[ \frac{y}{b} = 1 - \frac{x}{a} \implies y = b - \frac{b}{a}x \]
Thus, the slope \(m_1 = -\frac{b}{a}\).

**For the second line:**
\[ \frac{y}{b} = \frac{x}{a} - 1 \implies y = \frac{b}{a}x - b \]
Thus, the slope \(m_2 = \frac{b}{a}\).

### Step 2: Use the formula for the tangent of the angle between two lines
The formula for the tangent of the angle \(\theta\) between two lines with slopes \(m_1\) and \(m_2\) is given by:
\[ \tan \theta = \left|\frac{m_2 - m_1}{1 + m_1 m_2}\right| \]

### Step 3: Substitute the slopes into the formula
Substituting \(m_1\) and \(m_2\) into the formula:
\[ \tan \theta = \left|\frac{\frac{b}{a} - \left(-\frac{b}{a}\right)}{1 + \left(-\frac{b}{a}\right) \cdot \frac{b}{a}}\right| \]

### Step 4: Simplify the numerator
The numerator becomes:
\[ \frac{b}{a} + \frac{b}{a} = \frac{2b}{a} \]

### Step 5: Simplify the denominator
The denominator becomes:
\[ 1 - \frac{b^2}{a^2} = \frac{a^2 - b^2}{a^2} \]

### Step 6: Combine the results
Now substituting back into the formula:
\[ \tan \theta = \left|\frac{\frac{2b}{a}}{\frac{a^2 - b^2}{a^2}}\right| = \left|\frac{2b \cdot a^2}{a(a^2 - b^2)}\right| = \frac{2ab}{a^2 - b^2} \]

### Conclusion
Thus, we have shown that:
\[ \tan \theta = \frac{2ab}{a^2 - b^2} \]