If `theta` is an acute angle and `tantheta+cottheta=2,` find the value of `tan^7theta+cot^7theta` .

Options

A. 0
B. 1
C. 2
D. 3

Explanation

To solve the problem, we need to find the value of \( \tan^7 \theta + \cot^7 \theta \) given that \( \tan \theta + \cot \theta = 2 \).

### Step-by-Step Solution:

1. **Start with the given equation:**
\[ \tan \theta + \cot \theta = 2 \]
We know that \( \cot \theta = \frac{1}{\tan \theta} \). Therefore, we can rewrite the equation as:
\[ \tan \theta + \frac{1}{\tan \theta} = 2 \]

2. **Let \( x = \tan \theta \):**
Substituting \( x \) into the equation gives:
\[ x + \frac{1}{x} = 2 \]

3. **Multiply through by \( x \) to eliminate the fraction:**
\[ x^2 + 1 = 2x \]

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

5. **Recognize this as a perfect square:**
\[ (x - 1)^2 = 0 \]

6. **Solve for \( x \):**
\[ x - 1 = 0 \implies x = 1 \]
Thus, \( \tan \theta = 1 \).

7. **Find \( \theta \):**
Since \( \tan \theta = 1 \), we know that:
\[ \theta = \frac{\pi}{4} \]

8. **Calculate \( \tan^7 \theta + \cot^7 \theta \):**
Since \( \tan \theta = 1 \), we have:
\[ \tan^7 \theta = 1^7 = 1 \]
And since \( \cot \theta = \frac{1}{\tan \theta} = 1 \):
\[ \cot^7 \theta = 1^7 = 1 \]

9. **Combine the results:**
\[ \tan^7 \theta + \cot^7 \theta = 1 + 1 = 2 \]

### Final Answer:
\[ \tan^7 \theta + \cot^7 \theta = 2 \]