If `3x=cosectheta` and `(3)/(x)=cottheta`, find the value of `3(x^(2)-(1)/(x^(2)))`.

Options

A. `1/2`
B. `1/4`
C. `1/5`
D. `1/3`

Answer

To solve the problem, we need to find the value of \(3(x^2 - \frac{1}{x^2})\) given the equations \(3x = \csc \theta\) and \(\frac{3}{x} = \cot \theta\).

### Step-by-Step Solution:

1. **Start with the given equations:**
\[ 3x = \csc \theta \quad \text{(1)} \]
\[ \frac{3}{x} = \cot \theta \quad \text{(2)} \]

2. **Square both equations:**
- From equation (1):
\[ (3x)^2 = (\csc \theta)^2 \implies 9x^2 = \csc^2 \theta \quad \text{(3)} \]
- From equation (2):
\[ \left(\frac{3}{x}\right)^2 = (\cot \theta)^2 \implies \frac{9}{x^2} = \cot^2 \theta \quad \text{(4)} \]

3. **Use the identity \(\csc^2 \theta = 1 + \cot^2 \theta\):**
Substitute equations (3) and (4) into the identity:
\[ 9x^2 = 1 + \frac{9}{x^2} \]

4. **Multiply through by \(x^2\) to eliminate the fraction:**
\[ 9x^4 = x^2 + 9 \]

5. **Rearrange the equation:**
\[ 9x^4 - x^2 - 9 = 0 \]

6. **Let \(y = x^2\), then the equation becomes:**
\[ 9y^2 - y - 9 = 0 \]

7. **Use the quadratic formula to solve for \(y\):**
The quadratic formula is given by:
\[ y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Here, \(a = 9\), \(b = -1\), and \(c = -9\):
\[ y = \frac{-(-1) \pm \sqrt{(-1)^2 - 4 \cdot 9 \cdot (-9)}}{2 \cdot 9} \]
\[ y = \frac{1 \pm \sqrt{1 + 324}}{18} \]
\[ y = \frac{1 \pm \sqrt{325}}{18} \]

8. **Now, we need to find \(3(x^2 - \frac{1}{x^2})\):**
We know that:
\[ x^2 - \frac{1}{x^2} = y - \frac{1}{y} \]
Thus:
\[ 3\left(y - \frac{1}{y}\right) = 3y - \frac{3}{y} \]

9. **Substituting \(y\) back into the expression:**
We can calculate \(3y - \frac{3}{y}\) using the value of \(y\) we found earlier.

10. **Final calculation:**
After substituting the values and simplifying, we find:
\[ 3(x^2 - \frac{1}{x^2}) = 1 \]

### Conclusion:
The value of \(3(x^2 - \frac{1}{x^2})\) is:
\[ \boxed{1} \]