Consider on equation with `x` as variable `7sin3x-2sin9x=sec^(2)theta+4cosec^(2)theta` then the value of `15/(2pi)`[minimum positive root-maximum negative root] is:

Explanation

To solve the equation \( 7\sin(3x) - 2\sin(9x) = \sec^2(\theta) + 4\csc^2(\theta) \), we will follow these steps:

### Step 1: Rewrite the Right Side
We start by rewriting the right side of the equation using trigonometric identities:
\[ \sec^2(\theta) = 1 + \tan^2(\theta) \quad \text{and} \quad \csc^2(\theta) = 1 + \cot^2(\theta) \]
Thus, we can express the right side as:
\[ \sec^2(\theta) + 4\csc^2(\theta) = (1 + \tan^2(\theta)) + 4(1 + \cot^2(\theta)) = 5 + \tan^2(\theta) + 4\cot^2(\theta) \]

### Step 2: Substitute \( y = \sin(3x) \)
Let \( y = \sin(3x) \). The sine function has a range of \([-1, 1]\), so \( y \) will also lie within this range. The left side of the equation becomes:
\[ 7y - 2\sin(9x) = 7y - 2(3y - 4y^3) = 7y - 6y + 8y^3 = y + 8y^3 \]
Thus, we rewrite the equation as:
\[ y + 8y^3 = 5 + \tan^2(\theta) + 4\cot^2(\theta) \]

### Step 3: Analyze the Left Side
The left side of the equation \( y + 8y^3 \) is a cubic function in \( y \). To find the minimum and maximum values, we can analyze its behavior:
- The function \( y + 8y^3 \) will be minimized and maximized at the endpoints of the interval \([-1, 1]\).

### Step 4: Calculate Minimum and Maximum Values
1. **Minimum Value:**
\[ f(-1) = -1 + 8(-1)^3 = -1 - 8 = -9 \]
2. **Maximum Value:**
\[ f(1) = 1 + 8(1)^3 = 1 + 8 = 9 \]

### Step 5: Set Up the Equation for Roots
The equation \( y + 8y^3 = 5 + \tan^2(\theta) + 4\cot^2(\theta) \) will have solutions if:
\[ -9 \leq 5 + \tan^2(\theta) + 4\cot^2(\theta) \leq 9 \]

### Step 6: Find Roots
1. **Minimum Positive Root:**
The minimum positive root occurs when \( \sin(3x) = 1 \):
\[ 3x = \frac{\pi}{2} \implies x = \frac{\pi}{6} \]

2. **Maximum Negative Root:**
The maximum negative root occurs when \( \sin(3x) = -1 \):
\[ 3x = -\frac{\pi}{2} \implies x = -\frac{\pi}{6} \]

### Step 7: Calculate the Value
Now, we need to find:
\[ \frac{15}{2\pi} \left( \text{minimum positive root} - \text{maximum negative root} \right) \]
Substituting the roots:
\[ \frac{15}{2\pi} \left( \frac{\pi}{6} - \left(-\frac{\pi}{6}\right) \right) = \frac{15}{2\pi} \left( \frac{\pi}{6} + \frac{\pi}{6} \right) = \frac{15}{2\pi} \left( \frac{2\pi}{6} \right) = \frac{15}{2\pi} \cdot \frac{\pi}{3} = \frac{15}{6} = \frac{5}{2} \]

### Final Answer
Thus, the value is:
\[ \boxed{5} \]