If `(cosec theta - cot theta)=2`, the
`(cosec theta +cot theta)` is equal to

Options

A. 2
B. `1/2`
C. 1
D. `3/2`

Answer

To find the value of \( \csc \theta + \cot \theta \) given that \( \csc \theta - \cot \theta = 2 \), we can use algebraic identities.

### Step-by-step Solution:

1. **Let \( x = \csc \theta + \cot \theta \)** and \( y = \csc \theta - \cot \theta \). We know from the problem statement that:
\[ y = 2 \]

2. **Use the identity for the difference of squares**:
\[ (\csc \theta + \cot \theta)(\csc \theta - \cot \theta) = \csc^2 \theta - \cot^2 \theta \]
This can be rewritten as:
\[ x \cdot y = \csc^2 \theta - \cot^2 \theta \]

3. **Recall the Pythagorean identity**:
\[ \csc^2 \theta - \cot^2 \theta = 1 \]
Therefore, we can substitute this into our equation:
\[ x \cdot y = 1 \]

4. **Substituting the value of \( y \)**:
\[ x \cdot 2 = 1 \]

5. **Solve for \( x \)**:
\[ x = \frac{1}{2} \]

6. **Conclusion**:
Thus, we have:
\[ \csc \theta + \cot \theta = \frac{1}{2} \]

### Final Answer:
\[ \csc \theta + \cot \theta = \frac{1}{2} \]