If `cosectheta-cottheta =2`, find the value of `cosec^(2)theta+cot^(2)theta`.

Answer

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

### Step-by-Step Solution:

1. **Start with the given equation:**
\[ \csc \theta - \cot \theta = 2 \]

2. **Rationalize the expression:**
Multiply both sides by \( \csc \theta + \cot \theta \):
\[ (\csc \theta - \cot \theta)(\csc \theta + \cot \theta) = 2(\csc \theta + \cot \theta) \]

3. **Use the difference of squares:**
The left-hand side can be simplified using the identity \( a^2 - b^2 \):
\[ \csc^2 \theta - \cot^2 \theta = 2(\csc \theta + \cot \theta) \]

4. **Substitute the identity for \( \csc^2 \theta \):**
Recall that \( \csc^2 \theta = 1 + \cot^2 \theta \):
\[ (1 + \cot^2 \theta) - \cot^2 \theta = 2(\csc \theta + \cot \theta) \]
This simplifies to:
\[ 1 = 2(\csc \theta + \cot \theta) \]

5. **Solve for \( \csc \theta + \cot \theta \):**
Rearranging gives:
\[ \csc \theta + \cot \theta = \frac{1}{2} \]

6. **Now we have two equations:**
- \( \csc \theta - \cot \theta = 2 \) (Equation 1)
- \( \csc \theta + \cot \theta = \frac{1}{2} \) (Equation 2)

7. **Add the two equations:**
\[ (\csc \theta - \cot \theta) + (\csc \theta + \cot \theta) = 2 + \frac{1}{2} \]
This simplifies to:
\[ 2 \csc \theta = \frac{5}{2} \]

8. **Solve for \( \csc \theta \):**
\[ \csc \theta = \frac{5}{4} \]

9. **Find \( \cot \theta \):**
Use the identity \( \cot^2 \theta = \csc^2 \theta - 1 \):
\[ \cot^2 \theta = \left(\frac{5}{4}\right)^2 - 1 = \frac{25}{16} - 1 = \frac{25}{16} - \frac{16}{16} = \frac{9}{16} \]

10. **Now calculate \( \csc^2 \theta + \cot^2 \theta \):**
\[ \csc^2 \theta + \cot^2 \theta = \left(\frac{5}{4}\right)^2 + \frac{9}{16} = \frac{25}{16} + \frac{9}{16} = \frac{34}{16} = \frac{17}{8} \]

### Final Answer:
\[ \csc^2 \theta + \cot^2 \theta = \frac{17}{8} \]