Which of the following is not the quadratic equation whose roots are `cosec^2thetaand sec^2theta`?

Options

A. `x^2-6x+6=0`
B. `x^2-7x+7=0`
C. `x^2-4x+4=0`
D. None of these

Explanation

To determine which of the following is not a quadratic equation whose roots are \( \csc^2 \theta \) and \( \sec^2 \theta \), we will follow these steps:

### Step 1: Understand the Roots
The roots of the quadratic equation are given as \( \alpha = \csc^2 \theta \) and \( \beta = \sec^2 \theta \).

### Step 2: Calculate the Sum of the Roots
Using the formula for the sum of the roots of a quadratic equation:
\[ \alpha + \beta = \csc^2 \theta + \sec^2 \theta \]
We can express \( \csc^2 \theta \) and \( \sec^2 \theta \) in terms of sine and cosine:
\[ \csc^2 \theta = \frac{1}{\sin^2 \theta}, \quad \sec^2 \theta = \frac{1}{\cos^2 \theta} \]
Thus,
\[ \alpha + \beta = \frac{1}{\sin^2 \theta} + \frac{1}{\cos^2 \theta} \]

### Step 3: Find a Common Denominator
To add these fractions, we find a common denominator:
\[ \alpha + \beta = \frac{\cos^2 \theta + \sin^2 \theta}{\sin^2 \theta \cos^2 \theta} \]
Using the Pythagorean identity \( \sin^2 \theta + \cos^2 \theta = 1 \):
\[ \alpha + \beta = \frac{1}{\sin^2 \theta \cos^2 \theta} \]

### Step 4: Calculate the Product of the Roots
Using the formula for the product of the roots:
\[ \alpha \beta = \csc^2 \theta \cdot \sec^2 \theta \]
This can be expressed as:
\[ \alpha \beta = \frac{1}{\sin^2 \theta} \cdot \frac{1}{\cos^2 \theta} = \frac{1}{\sin^2 \theta \cos^2 \theta} \]

### Step 5: Form the Quadratic Equation
The general form of a quadratic equation is:
\[ ax^2 + bx + c = 0 \]
Using the relationships from the roots:
- Sum of the roots \( \alpha + \beta = -\frac{b}{a} \)
- Product of the roots \( \alpha \beta = \frac{c}{a} \)

From our calculations:
\[ -\frac{b}{a} = \frac{1}{\sin^2 \theta \cos^2 \theta}, \quad \frac{c}{a} = \frac{1}{\sin^2 \theta \cos^2 \theta} \]
This implies:
\[ -b = c \]

### Step 6: Analyze the Given Options
Now we need to check the options provided in the question to see which one does not satisfy the condition \( -b = c \).

1. **Option A**: \( b = -6 \), \( c = 6 \) → \( -(-6) = 6 \) (satisfies)
2. **Option B**: \( b = -7 \), \( c = 7 \) → \( -(-7) = 7 \) (satisfies)
3. **Option C**: \( b = -4 \), \( c = 4 \) → \( -(-4) = 4 \) (satisfies)
4. **Option D**: None of the above (implies that all options are valid)

### Conclusion
Since all options A, B, and C satisfy the condition \( -b = c \), the correct answer is:
**Option D: None of these** is the correct answer.