If `sec theta+tan theta=(1)/(2)`
The value of `sec theta=`

Answer

To solve the equation \( \sec \theta + \tan \theta = \frac{1}{2} \) and find the value of \( \sec \theta \), we can follow these steps:

### Step 1: Start with the given equation
We have:
\[ \sec \theta + \tan \theta = \frac{1}{2} \]

### Step 2: Multiply both sides by \( \sec \theta - \tan \theta \)
To simplify the equation, we multiply both sides by \( \sec \theta - \tan \theta \):
\[ (\sec \theta + \tan \theta)(\sec \theta - \tan \theta) = \frac{1}{2}(\sec \theta - \tan \theta) \]

### Step 3: Apply the difference of squares
Using the identity \( (a + b)(a - b) = a^2 - b^2 \), we get:
\[ \sec^2 \theta - \tan^2 \theta = \frac{1}{2}(\sec \theta - \tan \theta) \]

### Step 4: Use the Pythagorean identity
Recall the identity \( \sec^2 \theta - \tan^2 \theta = 1 \):
\[ 1 = \frac{1}{2}(\sec \theta - \tan \theta) \]

### Step 5: Multiply both sides by 2
To eliminate the fraction, multiply both sides by 2:
\[ 2 = \sec \theta - \tan \theta \]

### Step 6: Set up a system of equations
Now we have two equations:
1. \( \sec \theta + \tan \theta = \frac{1}{2} \)
2. \( \sec \theta - \tan \theta = 2 \)

### Step 7: Add the two equations
Adding these two equations:
\[ (\sec \theta + \tan \theta) + (\sec \theta - \tan \theta) = \frac{1}{2} + 2 \]
This simplifies to:
\[ 2 \sec \theta = \frac{1}{2} + 2 = \frac{1}{2} + \frac{4}{2} = \frac{5}{2} \]

### Step 8: Solve for \( \sec \theta \)
Now, divide both sides by 2:
\[ \sec \theta = \frac{5}{4} \]

### Final Answer
Thus, the value of \( \sec \theta \) is:
\[ \sec \theta = \frac{5}{4} \]