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

Answer

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

### Step 1: Multiply both sides by \( \sec \theta - \tan \theta \)

We start with the equation:
\[ \sec \theta + \tan \theta = \frac{1}{2} \]

Multiply both sides by \( \sec \theta - \tan \theta \):
\[ (\sec \theta + \tan \theta)(\sec \theta - \tan \theta) = \frac{1}{2}(\sec \theta - \tan \theta) \]

### Step 2: Simplify the left side using the identity

Using the identity \( \sec^2 \theta - \tan^2 \theta = 1 \), we can simplify the left side:
\[ \sec^2 \theta - \tan^2 \theta = 1 \]
So, we have:
\[ 1 = \frac{1}{2}(\sec \theta - \tan \theta) \]

### Step 3: Solve for \( \sec \theta - \tan \theta \)

Now, we can multiply both sides by 2:
\[ 2 = \sec \theta - \tan \theta \]

### Step 4: Set up a system of equations

Now we have two equations:
1. \( \sec \theta + \tan \theta = \frac{1}{2} \) (Equation 1)
2. \( \sec \theta - \tan \theta = 2 \) (Equation 2)

### Step 5: Add the two equations

Adding Equation 1 and Equation 2:
\[ (\sec \theta + \tan \theta) + (\sec \theta - \tan \theta) = \frac{1}{2} + 2 \]
This simplifies to:
\[ 2\sec \theta = \frac{1}{2} + 2 = \frac{1}{2} + \frac{4}{2} = \frac{5}{2} \]

### Step 6: Solve for \( \sec \theta \)

Now divide both sides by 2:
\[ \sec \theta = \frac{5}{4} \]

### Step 7: Substitute back to find \( \tan \theta \)

Now substitute \( \sec \theta \) back into Equation 1:
\[ \frac{5}{4} + \tan \theta = \frac{1}{2} \]
Rearranging gives:
\[ \tan \theta = \frac{1}{2} - \frac{5}{4} \]
Finding a common denominator:
\[ \tan \theta = \frac{2}{4} - \frac{5}{4} = -\frac{3}{4} \]

### Final Answer

Thus, the value of \( \tan \theta \) is:
\[ \boxed{-\frac{3}{4}} \]