If `sectheta+tantheta=2` , then the value of `sectheta` is

Options

A. `(4)/(5)`
B. 5
C. `(5)/(4)`
D. `sqrt(2)`

Answer

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

### Step 1: Use the Pythagorean Identity
We know from trigonometric identities that:
\[ \sec^2 \theta - \tan^2 \theta = 1 \]
This can be rewritten as:
\[ (\sec \theta - \tan \theta)(\sec \theta + \tan \theta) = 1 \]

### Step 2: Substitute the Given Value
We know from the problem that \( \sec \theta + \tan \theta = 2 \). We can substitute this into our equation:
\[ (\sec \theta - \tan \theta)(2) = 1 \]

### Step 3: Solve for \( \sec \theta - \tan \theta \)
From the equation above, we can isolate \( \sec \theta - \tan \theta \):
\[ \sec \theta - \tan \theta = \frac{1}{2} \]

### Step 4: Set Up the 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 5: Add the Two Equations
We can add Equation 1 and Equation 2:
\[ (\sec \theta + \tan \theta) + (\sec \theta - \tan \theta) = 2 + \frac{1}{2} \]
This simplifies to:
\[ 2 \sec \theta = 2 + \frac{1}{2} = \frac{5}{2} \]

### Step 6: Solve for \( \sec \theta \)
Now, divide both sides by 2:
\[ \sec \theta = \frac{5}{4} \]

### Conclusion
Thus, the value of \( \sec \theta \) is \( \frac{5}{4} \).

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