If `tan alpha` and `tan beta` are two solutions of `x^(2)-px +q = 0, cot alpha` and `cot beta` are the roots of `x^(2)- rx +s = 0` then the value of rs is equal to

Options

A. `(p)/(q^(2))`
B. `(q)/(p^(2))`
C. `(1)/(pq)`
D. pq

Answer

To solve the problem step by step, we will analyze the given quadratic equations and their roots.

### Step 1: Identify the roots of the first equation
The first equation is:
\[ x^2 - px + q = 0 \]
The roots of this equation are given as \( \tan \alpha \) and \( \tan \beta \).

Using Vieta's formulas:
- The sum of the roots \( \tan \alpha + \tan \beta = p \)
- The product of the roots \( \tan \alpha \tan \beta = q \)

### Step 2: Identify the roots of the second equation
The second equation is:
\[ x^2 - rx + s = 0 \]
The roots of this equation are \( \cot \alpha \) and \( \cot \beta \).

Again, using Vieta's formulas:
- The sum of the roots \( \cot \alpha + \cot \beta = r \)
- The product of the roots \( \cot \alpha \cot \beta = s \)

### Step 3: Express \( r \) in terms of \( p \) and \( q \)
We know that:
\[ \cot \alpha = \frac{1}{\tan \alpha} \quad \text{and} \quad \cot \beta = \frac{1}{\tan \beta} \]

Thus, the sum of the roots can be expressed as:
\[ r = \cot \alpha + \cot \beta = \frac{1}{\tan \alpha} + \frac{1}{\tan \beta} \]
This can be rewritten using the formula for the sum of fractions:
\[ r = \frac{\tan \beta + \tan \alpha}{\tan \alpha \tan \beta} \]
Substituting the values from Step 1:
\[ r = \frac{p}{q} \]

### Step 4: Express \( s \) in terms of \( q \)
Now, for the product of the roots:
\[ s = \cot \alpha \cot \beta = \frac{1}{\tan \alpha} \cdot \frac{1}{\tan \beta} = \frac{1}{\tan \alpha \tan \beta} \]
Substituting the value from Step 1:
\[ s = \frac{1}{q} \]

### Step 5: Calculate \( rs \)
Now we need to find the product \( rs \):
\[ rs = r \cdot s = \left(\frac{p}{q}\right) \cdot \left(\frac{1}{q}\right) \]
This simplifies to:
\[ rs = \frac{p}{q^2} \]

### Final Answer
Thus, the value of \( rs \) is:
\[ \boxed{\frac{p}{q^2}} \]