If `sec theta + tan theta = p`, `(p != 0)` then `sec theta` is equal to

Options

A. `(p - 1/p),
B. `2 (p - 1/p),
C. `(p + 1/p),
D. `1/2 (p + 1/p),

Answer

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

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

2. **Using the Identity**:
We know that:
\[ \sec^2 \theta - \tan^2 \theta = 1 \]
This can be factored as:
\[ (\sec \theta + \tan \theta)(\sec \theta - \tan \theta) = 1 \]
Since we know that \(\sec \theta + \tan \theta = p\), we can substitute \(p\) into the equation:
\[ p(\sec \theta - \tan \theta) = 1 \]

3. **Solving for \(\sec \theta - \tan \theta\)**:
Rearranging gives us:
\[ \sec \theta - \tan \theta = \frac{1}{p} \]

4. **Setting Up the Equations**:
Now we have two equations:
- Equation 1: \(\sec \theta + \tan \theta = p\)
- Equation 2: \(\sec \theta - \tan \theta = \frac{1}{p}\)

5. **Adding the Two Equations**:
Adding Equation 1 and Equation 2:
\[ (\sec \theta + \tan \theta) + (\sec \theta - \tan \theta) = p + \frac{1}{p} \]
This simplifies to:
\[ 2\sec \theta = p + \frac{1}{p} \]

6. **Solving for \(\sec \theta\)**:
Dividing both sides by 2 gives us:
\[ \sec \theta = \frac{p + \frac{1}{p}}{2} \]

7. **Final Result**:
Therefore, the value of \(\sec \theta\) is:
\[ \sec \theta = \frac{p + \frac{1}{p}}{2} \]