If `alpha and beta (alpha gt beta)` are the roots of `x^(2) + kx - 1 =0`, then find the value of `tan^(-1) alpha - tan^(-1) beta`

Explanation

To solve the problem of finding the value of \( \tan^{-1} \alpha - \tan^{-1} \beta \) where \( \alpha \) and \( \beta \) are the roots of the equation \( x^2 + kx - 1 = 0 \) and \( \alpha > \beta \), we can follow these steps:

### Step 1: Identify the roots using the quadratic formula
The roots of the quadratic equation \( ax^2 + bx + c = 0 \) can be found using the quadratic formula:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
For our equation \( x^2 + kx - 1 = 0 \), we have \( a = 1 \), \( b = k \), and \( c = -1 \). Thus, the roots are:
\[ \alpha, \beta = \frac{-k \pm \sqrt{k^2 + 4}}{2} \]

### Step 2: Calculate the sum and product of the roots
From Vieta's formulas, we know:
- The sum of the roots \( \alpha + \beta = -\frac{b}{a} = -k \)
- The product of the roots \( \alpha \beta = \frac{c}{a} = -1 \)

### Step 3: Use the identity for the difference of inverse tangents
We can use the identity for the difference of inverse tangents:
\[ \tan^{-1} \alpha - \tan^{-1} \beta = \tan^{-1} \left( \frac{\alpha - \beta}{1 + \alpha \beta} \right) \]

### Step 4: Substitute the values of \( \alpha \) and \( \beta \)
From the product of the roots, we know \( \alpha \beta = -1 \). Therefore, the expression simplifies to:
\[ \tan^{-1} \left( \frac{\alpha - \beta}{1 - 1} \right) \]
This results in a division by zero, indicating that we need to analyze the limit.

### Step 5: Analyze the limit
Since \( \alpha > \beta \), \( \alpha - \beta \) is positive, and as \( \alpha \) approaches \( \beta \), the expression tends toward infinity. Thus:
\[ \tan^{-1} \left( \frac{\alpha - \beta}{0} \right) \to \tan^{-1}(\infty) = \frac{\pi}{2} \]

### Conclusion
Therefore, the value of \( \tan^{-1} \alpha - \tan^{-1} \beta \) is:
\[ \frac{\pi}{2} \]