If `6x=sectheta` and `(6)/(x)=tantheta`, find the value of `9(x^(2)-(1)/(x^(2)))`.

Options

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

Explanation

To solve the problem step by step, we start with the given equations:

1. **Given Equations:**
- \( 6x = \sec \theta \) (Equation 1)
- \( \frac{6}{x} = \tan \theta \) (Equation 2)

2. **Add the Two Equations:**
- Adding Equation 1 and Equation 2:
\[ 6x + \frac{6}{x} = \sec \theta + \tan \theta \]
- Factor out 6 from the left side:
\[ 6 \left( x + \frac{1}{x} \right) = \sec \theta + \tan \theta \]
- Let this be Equation 3:
\[ 6 \left( x + \frac{1}{x} \right) = \sec \theta + \tan \theta \quad \text{(Equation 3)} \]

3. **Subtract the Two Equations:**
- Now, subtract Equation 2 from Equation 1:
\[ 6x - \frac{6}{x} = \sec \theta - \tan \theta \]
- Factor out 6 from the left side:
\[ 6 \left( x - \frac{1}{x} \right) = \sec \theta - \tan \theta \]
- Let this be Equation 4:
\[ 6 \left( x - \frac{1}{x} \right) = \sec \theta - \tan \theta \quad \text{(Equation 4)} \]

4. **Multiply Equations 3 and 4:**
- Now, multiply Equation 3 and Equation 4:
\[ \left( 6 \left( x + \frac{1}{x} \right) \right) \left( 6 \left( x - \frac{1}{x} \right) \right) = \left( \sec \theta + \tan \theta \right) \left( \sec \theta - \tan \theta \right) \]
- The left side becomes:
\[ 36 \left( x^2 - \frac{1}{x^2} \right) \]
- The right side uses the identity \( a^2 - b^2 = (a + b)(a - b) \):
\[ \sec^2 \theta - \tan^2 \theta = 1 \]
- Thus, we have:
\[ 36 \left( x^2 - \frac{1}{x^2} \right) = 1 \]

5. **Solve for \( x^2 - \frac{1}{x^2} \):**
- Divide both sides by 36:
\[ x^2 - \frac{1}{x^2} = \frac{1}{36} \]

6. **Find \( 9 \left( x^2 - \frac{1}{x^2} \right) \):**
- Now, multiply both sides by 9:
\[ 9 \left( x^2 - \frac{1}{x^2} \right) = 9 \cdot \frac{1}{36} = \frac{9}{36} = \frac{1}{4} \]

7. **Final Answer:**
- Therefore, the value of \( 9 \left( x^2 - \frac{1}{x^2} \right) \) is:
\[ \frac{1}{4} \]