If `theta` is an acute angle and `tan theta + cot theta=2`, then the value of `tan^(5)theta + cot^(10) theta` is:

Options

A. 1
B. 2
C. 3
D. 4

Answer

To solve the problem step by step, we start with the given equation:

1. **Given Equation**:
\[ \tan \theta + \cot \theta = 2 \]

2. **Understanding the Relationship**:
We know that:
\[ \cot \theta = \frac{1}{\tan \theta} \]
Therefore, we can rewrite the equation as:
\[ \tan \theta + \frac{1}{\tan \theta} = 2 \]

3. **Letting \( x = \tan \theta \)**:
This gives us:
\[ x + \frac{1}{x} = 2 \]

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

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

6. **Factoring the Quadratic**:
This can be factored as:
\[ (x - 1)^2 = 0 \]

7. **Finding the Value of \( x \)**:
Solving for \( x \) gives:
\[ x - 1 = 0 \implies x = 1 \]
Hence, we have:
\[ \tan \theta = 1 \]

8. **Finding \( \cot \theta \)**:
Since \( \tan \theta = 1 \):
\[ \cot \theta = \frac{1}{\tan \theta} = 1 \]

9. **Calculating \( \tan^5 \theta + \cot^{10} \theta \)**:
Now we substitute back to find:
\[ \tan^5 \theta + \cot^{10} \theta = 1^5 + 1^{10} = 1 + 1 = 2 \]

10. **Final Answer**:
Therefore, the value of \( \tan^5 \theta + \cot^{10} \theta \) is:
\[ \boxed{2} \]