If `sec theta + tan theta = 12.5`, then `sec theta - tan theta = ?`

Options

A. `0.4`
B. 1
C. `0.8`
D. `0.08`

Answer

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

1. **Given Equation**:
\[ \sec \theta + \tan \theta = 12.5 \]

2. **Let**:
\[ x = \sec \theta - \tan \theta \]

3. **Multiply the two equations**:
\[ (\sec \theta + \tan \theta)(\sec \theta - \tan \theta) = 12.5 \cdot x \]

4. **Use the identity**:
\[ A + B \cdot A - B = A^2 - B^2 \]
Here, \( A = \sec \theta \) and \( B = \tan \theta \):
\[ \sec^2 \theta - \tan^2 \theta = 12.5x \]

5. **Use the Pythagorean identity**:
\[ \sec^2 \theta - \tan^2 \theta = 1 \]
So we can rewrite the equation as:
\[ 1 = 12.5x \]

6. **Solve for \( x \)**:
\[ x = \frac{1}{12.5} \]

7. **Convert to decimal**:
\[ x = \frac{1}{12.5} = \frac{1 \times 2}{12.5 \times 2} = \frac{2}{25} = 0.08 \]

8. **Final Result**:
\[ \sec \theta - \tan \theta = 0.08 \]