The minimum value of `atan^2x+bcot^2x` equals the maximum value of `asin^2theta+bcos^2theta` where `a > b > 0.` The `a/b` is 2 (b) 4 (c) 6 (d) 8

Explanation

To solve the problem, we need to find the relationship between \( a \) and \( b \) given the conditions of the minimum and maximum values of the trigonometric functions involved.

### Step-by-Step Solution:

1. **Understanding the Minimum Value of \( a \tan^2 x + b \cot^2 x \)**:
- The expression \( a \tan^2 x + b \cot^2 x \) can be rewritten using the identity \( \cot^2 x = \frac{1}{\tan^2 x} \).
- Let \( t = \tan^2 x \). Then, the expression becomes \( a t + \frac{b}{t} \).

2. **Finding the Minimum Value**:
- To find the minimum value of \( a t + \frac{b}{t} \), we can use the AM-GM inequality:
\[ a t + \frac{b}{t} \geq 2 \sqrt{a \cdot \frac{b}{t}} = 2\sqrt{ab} \]
- The equality holds when \( at = \frac{b}{t} \) or \( t^2 = \frac{b}{a} \), leading to \( t = \sqrt{\frac{b}{a}} \).

3. **Setting the Minimum Value**:
- Thus, the minimum value of \( a \tan^2 x + b \cot^2 x \) is \( 2\sqrt{ab} \).

4. **Understanding the Maximum Value of \( a \sin^2 \theta + b \cos^2 \theta \)**:
- The expression \( a \sin^2 \theta + b \cos^2 \theta \) can be rewritten using the identity \( \cos^2 \theta = 1 - \sin^2 \theta \):
\[ a \sin^2 \theta + b (1 - \sin^2 \theta) = (a - b) \sin^2 \theta + b \]

5. **Finding the Maximum Value**:
- The maximum value occurs when \( \sin^2 \theta \) is maximized. If \( a > b \), the maximum value occurs at \( \sin^2 \theta = 1 \):
\[ \text{Maximum value} = a - b + b = a \]

6. **Equating the Minimum and Maximum Values**:
- From the problem, we have:
\[ 2\sqrt{ab} = a \]

7. **Squaring Both Sides**:
- Squaring both sides gives:
\[ 4ab = a^2 \]

8. **Rearranging the Equation**:
- Rearranging gives:
\[ a^2 - 4ab = 0 \]
- Factoring out \( a \):
\[ a(a - 4b) = 0 \]
- Since \( a > 0 \), we have:
\[ a - 4b = 0 \implies a = 4b \]

9. **Finding \( \frac{a}{b} \)**:
- Therefore:
\[ \frac{a}{b} = 4 \]

### Final Answer:
The value of \( \frac{a}{b} \) is \( 4 \).