If ` sin theta+cosec theta=2, "then the volume of" sin ^(10)theta+cosec^(10)theta, is`

Answer

To solve the problem, we need to find the value of \( \sin^{10} \theta + \csc^{10} \theta \) given that \( \sin \theta + \csc \theta = 2 \).

### Step-by-Step Solution:

1. **Start with the given equation:**
\[ \sin \theta + \csc \theta = 2 \]
Recall that \( \csc \theta = \frac{1}{\sin \theta} \).

2. **Substituting for \( \csc \theta \):**
\[ \sin \theta + \frac{1}{\sin \theta} = 2 \]

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

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

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

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

7. **Finding \( \csc \theta \):**
Since \( \sin \theta = 1 \),
\[ \csc \theta = \frac{1}{\sin \theta} = \frac{1}{1} = 1 \]

8. **Calculating \( \sin^{10} \theta + \csc^{10} \theta \):**
\[ \sin^{10} \theta + \csc^{10} \theta = 1^{10} + 1^{10} = 1 + 1 = 2 \]

### Final Answer:
The value of \( \sin^{10} \theta + \csc^{10} \theta \) is \( 2 \).

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