If `theta ` is a positive acute angle and `tan theta + cot theta = 2`, then the value of sec `theta ` is

Options

A. `(1)/( sqrt(2))`
B. `sqrt(2)`
C. `1`
D. `0`

Answer

To solve the problem, we need to find the value of sec θ given that tan θ + cot θ = 2 and θ is a positive acute angle.

### Step-by-step Solution:

1. **Understanding the Equation**:
We start with the equation given in the problem:
\[ \tan \theta + \cot \theta = 2 \]
Recall that:
\[ \cot \theta = \frac{1}{\tan \theta} \]
Therefore, we can rewrite the equation as:
\[ \tan \theta + \frac{1}{\tan \theta} = 2 \]

2. **Letting \( x = \tan \theta \)**:
To simplify the equation, let:
\[ x = \tan \theta \]
Then the equation becomes:
\[ x + \frac{1}{x} = 2 \]

3. **Multiplying through by \( x \)**:
To eliminate the fraction, multiply both sides by \( x \):
\[ x^2 + 1 = 2x \]

4. **Rearranging the Equation**:
Rearranging gives us a standard quadratic equation:
\[ x^2 - 2x + 1 = 0 \]

5. **Factoring the Quadratic**:
This can be factored as:
\[ (x - 1)^2 = 0 \]
Thus, we find:
\[ x - 1 = 0 \quad \Rightarrow \quad x = 1 \]

6. **Finding \( \tan \theta \)**:
Since \( x = \tan \theta \), we have:
\[ \tan \theta = 1 \]

7. **Determining \( \theta \)**:
The angle \( \theta \) that satisfies \( \tan \theta = 1 \) in the range of acute angles is:
\[ \theta = 45^\circ \]

8. **Finding \( \sec \theta \)**:
Now, we need to find \( \sec \theta \):
\[ \sec \theta = \frac{1}{\cos \theta} \]
For \( \theta = 45^\circ \):
\[ \cos 45^\circ = \frac{1}{\sqrt{2}} \quad \Rightarrow \quad \sec 45^\circ = \frac{1}{\frac{1}{\sqrt{2}}} = \sqrt{2} \]

### Final Answer:
Thus, the value of \( \sec \theta \) is:
\[ \sec \theta = \sqrt{2} \]

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