If `cos theta + sec theta = 2` then `cos^(n)theta+ sec^(n)theta` is equal to ___

Options

A. `2^(n)`
B. 2
C. `(-2)^(n)`
D. none of these

Answer

To solve the problem, we need to find the value of \( \cos^n \theta + \sec^n \theta \) given that \( \cos \theta + \sec \theta = 2 \).

### Step-by-Step Solution:

1. **Start with the given equation:**
\[ \cos \theta + \sec \theta = 2 \]

2. **Recall that \( \sec \theta = \frac{1}{\cos \theta} \):**
\[ \cos \theta + \frac{1}{\cos \theta} = 2 \]

3. **Multiply both sides by \( \cos \theta \) (assuming \( \cos \theta \neq 0 \)):**
\[ \cos^2 \theta + 1 = 2 \cos \theta \]

4. **Rearrange the equation:**
\[ \cos^2 \theta - 2 \cos \theta + 1 = 0 \]

5. **Recognize that this is a perfect square:**
\[ (\cos \theta - 1)^2 = 0 \]

6. **Solve for \( \cos \theta \):**
\[ \cos \theta - 1 = 0 \implies \cos \theta = 1 \]

7. **Substituting \( \cos \theta = 1 \) to find \( \sec \theta \):**
\[ \sec \theta = \frac{1}{\cos \theta} = \frac{1}{1} = 1 \]

8. **Now calculate \( \cos^n \theta + \sec^n \theta \):**
\[ \cos^n \theta + \sec^n \theta = 1^n + 1^n = 1 + 1 = 2 \]

Thus, we conclude that:
\[ \cos^n \theta + \sec^n \theta = 2 \]

### Final Answer:
The value of \( \cos^n \theta + \sec^n \theta \) is \( 2 \).