If `sec theta + tan theta =2`, what is the value of `sec theta` ?

Options

A. `3/2`
B. `sqrt2`
C. `5/2`
D. `5/4`

Answer

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

### Step 1: Write down the given equation
We start with the equation:
\[ \sec \theta + \tan \theta = 2 \]

### Step 2: Use the identity for secant and tangent
We know the identity:
\[ \sec^2 \theta - \tan^2 \theta = 1 \]
This can be factored as:
\[ (\sec \theta + \tan \theta)(\sec \theta - \tan \theta) = 1 \]

### Step 3: Substitute the known value
From the given equation, we can substitute \( \sec \theta + \tan \theta = 2 \):
\[ 2(\sec \theta - \tan \theta) = 1 \]

### Step 4: Solve for \( \sec \theta - \tan \theta \)
Now, we can solve for \( \sec \theta - \tan \theta \):
\[ \sec \theta - \tan \theta = \frac{1}{2} \]

### Step 5: Set up a system of equations
Now we have two equations:
1. \( \sec \theta + \tan \theta = 2 \) (Equation 1)
2. \( \sec \theta - \tan \theta = \frac{1}{2} \) (Equation 2)

### Step 6: Add the two equations
Adding Equation 1 and Equation 2:
\[ (\sec \theta + \tan \theta) + (\sec \theta - \tan \theta) = 2 + \frac{1}{2} \]
This simplifies to:
\[ 2\sec \theta = 2 + \frac{1}{2} = \frac{4}{2} + \frac{1}{2} = \frac{5}{2} \]

### Step 7: Solve for \( \sec \theta \)
Now, divide both sides by 2:
\[ \sec \theta = \frac{5}{4} \]

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

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