If : `tan theta+cot theta=2, "then" : tan^(2)theta-tan^(3)theta=`

Options

A. `cot^(3)theta-cot^(2)theta`
B. 1
C. `cot^(2)theta+cot^(3)theta`
D. 2

Answer

To solve the problem, we need to find the value of \( \tan^2 \theta - \tan^3 \theta \) given that \( \tan \theta + \cot \theta = 2 \).

### Step-by-Step Solution:

1. **Start with the given equation:**
\[ \tan \theta + \cot \theta = 2 \]
Recall that \( \cot \theta = \frac{1}{\tan \theta} \).

2. **Substitute \( \cot \theta \) in the equation:**
\[ \tan \theta + \frac{1}{\tan \theta} = 2 \]

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

4. **Rearrange the equation to form a quadratic equation:**
\[ \tan^2 \theta - 2 \tan \theta + 1 = 0 \]

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

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

7. **Determine the angle \( \theta \):**
\[ \theta = \frac{\pi}{4} \quad \text{(or 45 degrees)} \]

8. **Now calculate \( \tan^2 \theta - \tan^3 \theta \):**
\[ \tan^2 \theta - \tan^3 \theta = \tan^2 \left(\frac{\pi}{4}\right) - \tan^3 \left(\frac{\pi}{4}\right) \]

9. **Substituting the value of \( \tan \left(\frac{\pi}{4}\right) \):**
\[ = 1^2 - 1^3 = 1 - 1 = 0 \]

### Final Answer:
\[ \tan^2 \theta - \tan^3 \theta = 0 \]