If `sintheta+cosectheta=2`, the value of `sin^(100)theta+cosec^(100)theta` is :

Options

A. 1
B. 2
C. 3
D. 4

Answer

To solve the problem, we need to find the value of \( \sin^{100} \theta + \csc^{100} \theta \) given that \( \sin \theta + \csc \theta = 2 \).

### Step-by-Step Solution:

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

2. **Rewrite \( \csc \theta \) in terms of \( \sin \theta \):**
\[ \csc \theta = \frac{1}{\sin \theta} \]
So, we can rewrite the equation as:
\[ \sin \theta + \frac{1}{\sin \theta} = 2 \]

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

4. **Rearranging the equation gives:**
\[ \sin^2 \theta - 2 \sin \theta + 1 = 0 \]

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

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

7. **Determine the angle \( \theta \):**
The value of \( \theta \) for which \( \sin \theta = 1 \) is:
\[ \theta = 90^\circ \]

8. **Now find \( \sin^{100} \theta + \csc^{100} \theta \):**
Since \( \sin 90^\circ = 1 \):
\[ \sin^{100} 90^\circ = 1^{100} = 1 \]
And since \( \csc 90^\circ = 1 \):
\[ \csc^{100} 90^\circ = 1^{100} = 1 \]

9. **Combine the results:**
\[ \sin^{100} \theta + \csc^{100} \theta = 1 + 1 = 2 \]

### Final Answer:
\[ \sin^{100} \theta + \csc^{100} \theta = 2 \]