If sin A + cosec A = 2,
find the value of `sin^2 A + "cosec"^2 A`.

Answer

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

### Step-by-Step Solution:

1. **Start with the given equation:**
\[ \sin A + \csc A = 2 \]

2. **Substitute \( \csc A \) with \( \frac{1}{\sin A} \):**
\[ \sin A + \frac{1}{\sin A} = 2 \]

3. **Multiply both sides by \( \sin A \) to eliminate the fraction:**
\[ \sin^2 A + 1 = 2 \sin A \]

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

5. **Recognize that the left side is a perfect square:**
\[ (\sin A - 1)^2 = 0 \]

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

7. **Find \( \csc A \):**
\[ \csc A = \frac{1}{\sin A} = \frac{1}{1} = 1 \]

8. **Now calculate \( \sin^2 A + \csc^2 A \):**
\[ \sin^2 A + \csc^2 A = \sin^2 A + \left(\frac{1}{\sin A}\right)^2 \]
\[ = 1^2 + 1^2 = 1 + 1 = 2 \]

### Final Answer:
\[ \sin^2 A + \csc^2 A = 2 \]