If `4x = sec theta` and `(4)/(x) = tan theta` then `8 (x^(2) - (1)/(x^(2)))` is

Options

A. `1/16`
B. `1/8`
C. `1/2`
D. `1/4`

Answer

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

1. **Given Equations**:
\[ 4x = \sec \theta \]
\[ \frac{4}{x} = \tan \theta \]

2. **Using the Identity**:
We know the trigonometric identity:
\[ \sec^2 \theta - \tan^2 \theta = 1 \]
We will substitute the values of \(\sec \theta\) and \(\tan \theta\) from the equations above.

3. **Substituting Values**:
Substitute \(\sec \theta = 4x\) and \(\tan \theta = \frac{4}{x}\) into the identity:
\[ (4x)^2 - \left(\frac{4}{x}\right)^2 = 1 \]

4. **Expanding the Squares**:
Calculate the squares:
\[ 16x^2 - \frac{16}{x^2} = 1 \]

5. **Rearranging the Equation**:
To isolate \(x^2 - \frac{1}{x^2}\), we can rearrange the equation:
\[ 16x^2 - \frac{16}{x^2} = 1 \]
Factor out 16:
\[ 16\left(x^2 - \frac{1}{x^2}\right) = 1 \]

6. **Dividing by 16**:
Divide both sides by 16:
\[ x^2 - \frac{1}{x^2} = \frac{1}{16} \]

7. **Finding \(8\left(x^2 - \frac{1}{x^2}\right)\)**:
Now, we need to find \(8\left(x^2 - \frac{1}{x^2}\right)\):
\[ 8\left(x^2 - \frac{1}{x^2}\right) = 8 \cdot \frac{1}{16} = \frac{8}{16} = \frac{1}{2} \]

8. **Final Result**:
Thus, the value of \(8\left(x^2 - \frac{1}{x^2}\right)\) is:
\[ \frac{1}{2} \]