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

Options

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

Answer

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

1. \( 4x = \sec \theta \)
2. \( \frac{4}{x} = \tan \theta \)

We need to find the value of \( 8\left(x^2 - \frac{1}{x^2}\right) \).

### Step 1: Express \( x \) in terms of \( \sec \theta \)

From the first equation, we can express \( x \) as:
\[ x = \frac{\sec \theta}{4} \]

**Hint**: To isolate \( x \), divide both sides of the equation by 4.

### Step 2: Express \( \frac{1}{x} \)

Now, we can find \( \frac{1}{x} \):
\[ \frac{1}{x} = \frac{4}{\sec \theta} \]

**Hint**: To find \( \frac{1}{x} \), take the reciprocal of \( x \).

### Step 3: Substitute \( \tan \theta \)

From the second equation, we can express \( \tan \theta \) as:
\[ \tan \theta = \frac{4}{x} \]

Substituting \( x = \frac{\sec \theta}{4} \) into this equation gives:
\[ \tan \theta = \frac{4}{\frac{\sec \theta}{4}} = \frac{16}{\sec \theta} \]

**Hint**: Use the definition of \( \tan \theta \) and substitute the value of \( x \).

### Step 4: Use the Pythagorean identity

Recall that:
\[ \sec^2 \theta = 1 + \tan^2 \theta \]

Substituting \( \tan \theta = \frac{16}{\sec \theta} \) into the identity:
\[ \sec^2 \theta = 1 + \left(\frac{16}{\sec \theta}\right)^2 \]

Multiplying through by \( \sec^2 \theta \):
\[ \sec^4 \theta = \sec^2 \theta + 256 \]

**Hint**: This step uses the relationship between secant and tangent.

### Step 5: Solve for \( x^2 \) and \( \frac{1}{x^2} \)

Now, we can find \( x^2 \) and \( \frac{1}{x^2} \):
\[ x^2 = \left(\frac{\sec \theta}{4}\right)^2 = \frac{\sec^2 \theta}{16} \]
\[ \frac{1}{x^2} = \left(\frac{4}{\sec \theta}\right)^2 = \frac{16}{\sec^2 \theta} \]

**Hint**: Square both sides of the equations for \( x \) and \( \frac{1}{x} \).

### Step 6: Substitute into the expression

Now substitute \( x^2 \) and \( \frac{1}{x^2} \) into the expression \( 8\left(x^2 - \frac{1}{x^2}\right) \):
\[ 8\left(x^2 - \frac{1}{x^2}\right) = 8\left(\frac{\sec^2 \theta}{16} - \frac{16}{\sec^2 \theta}\right) \]

### Step 7: Simplify the expression

This simplifies to:
\[ = 8\left(\frac{\sec^4 \theta - 16}{16\sec^2 \theta}\right) = \frac{8(\sec^4 \theta - 16)}{16\sec^2 \theta} = \frac{\sec^4 \theta - 16}{2\sec^2 \theta} \]

### Step 8: Use the Pythagorean identity again

Using the identity \( \sec^4 \theta = \sec^2 \theta + 256 \):
\[ \frac{(\sec^2 \theta + 256) - 16}{2\sec^2 \theta} = \frac{\sec^2 \theta + 240}{2\sec^2 \theta} \]

### Final Step: Evaluate the expression

This gives us:
\[ = \frac{240}{2\sec^2 \theta} = \frac{120}{\sec^2 \theta} \]

Now, since \( \sec^2 \theta - \tan^2 \theta = 1 \), we can conclude that the value of \( 8\left(x^2 - \frac{1}{x^2}\right) \) simplifies to \( 8 \).

Thus, the final answer is:
\[ \boxed{8} \]