If `sintheta+cosec theta=2`, then `sin^(2)theta+cosec ^(2)theta` is equal to

Options

A. `1`
B. `4`
C. `2`
D. None of these

Explanation

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

### Step-by-Step Solution:

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

2. **Square both sides:**
\[ (\sin \theta + \csc \theta)^2 = 2^2 \]
This gives us:
\[ \sin^2 \theta + 2 \sin \theta \csc \theta + \csc^2 \theta = 4 \]

3. **Substitute \( \csc \theta \) with \( \frac{1}{\sin \theta} \):**
\[ \sin^2 \theta + 2 \sin \theta \left(\frac{1}{\sin \theta}\right) + \left(\frac{1}{\sin \theta}\right)^2 = 4 \]
This simplifies to:
\[ \sin^2 \theta + 2 + \frac{1}{\sin^2 \theta} = 4 \]

4. **Rearranging the equation:**
\[ \sin^2 \theta + \frac{1}{\sin^2 \theta} = 4 - 2 \]
Thus:
\[ \sin^2 \theta + \frac{1}{\sin^2 \theta} = 2 \]

5. **Recognizing that \( \csc^2 \theta = \frac{1}{\sin^2 \theta} \):**
Therefore, we can rewrite the equation as:
\[ \sin^2 \theta + \csc^2 \theta = 2 \]

### Final Result:
Thus, the value of \( \sin^2 \theta + \csc^2 \theta \) is \( 2 \).