If `cot alpha and cot beta` are the roots of the equation `x^(2) + bx + c = 0` with `b!=0` then the value of `cot (alpha + beta)` is :

Options

A. `(c -1)/(b)`
B. `(1 -c)/(b)`
C. `(c )/(c - 1)`
D. `(b)/(1 - c)`

Answer

To find the value of \( \cot(\alpha + \beta) \) given that \( \cot \alpha \) and \( \cot \beta \) are the roots of the quadratic equation \( x^2 + bx + c = 0 \), we can follow these steps:

### Step 1: Identify the roots
The roots of the equation \( x^2 + bx + c = 0 \) are given as \( \cot \alpha \) and \( \cot \beta \).

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

### Step 3: Use the cotangent addition formula
We can express \( \cot(\alpha + \beta) \) using the cotangent addition formula:
\[ \cot(\alpha + \beta) = \frac{\cot \alpha \cdot \cot \beta - 1}{\cot \alpha + \cot \beta} \]

### Step 4: Substitute the values
Now, substituting the values we found from Vieta's formulas:
- \( \cot \alpha \cdot \cot \beta = c \)
- \( \cot \alpha + \cot \beta = -b \)

So, we have:
\[ \cot(\alpha + \beta) = \frac{c - 1}{-b} \]

### Step 5: Final expression
Thus, the value of \( \cot(\alpha + \beta) \) is:
\[ \cot(\alpha + \beta) = -\frac{c - 1}{b} \]

### Summary
The final answer is:
\[ \cot(\alpha + \beta) = -\frac{c - 1}{b} \]

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