If ` sec theta + tan theta =x, " find the value of " sec theta. `

Explanation

To solve the problem where we need to find the value of \( \sec \theta \) given that \( \sec \theta + \tan \theta = x \), we can follow these steps:

### Step-by-Step Solution:

1. **Start with the given equation:**
\[ \sec \theta + \tan \theta = x \]

2. **Use the property of secant and tangent:**
According to the property, if \( \sec \theta + \tan \theta = a \), then:
\[ \sec \theta - \tan \theta = \frac{1}{a} \]
In our case, substituting \( a \) with \( x \):
\[ \sec \theta - \tan \theta = \frac{1}{x} \]

3. **Now we have two equations:**
\[ \sec \theta + \tan \theta = x \quad (1) \]
\[ \sec \theta - \tan \theta = \frac{1}{x} \quad (2) \]

4. **Add the two equations (1) and (2):**
\[ (\sec \theta + \tan \theta) + (\sec \theta - \tan \theta) = x + \frac{1}{x} \]
This simplifies to:
\[ 2 \sec \theta = x + \frac{1}{x} \]

5. **Solve for \( \sec \theta \):**
Divide both sides by 2:
\[ \sec \theta = \frac{x + \frac{1}{x}}{2} \]

6. **Simplify the expression:**
\[ \sec \theta = \frac{x + \frac{1}{x}}{2} = \frac{x^2 + 1}{2x} \]

### Final Answer:
\[ \sec \theta = \frac{x + \frac{1}{x}}{2} \]