If `0ltthetaltpi`, then the minimum value of `sin^(5)theta+"cosec"^(5)theta,` is

Options

A. 0
B. 1
C. 2
D. none of these

Answer

To find the minimum value of \( \sin^5 \theta + \csc^5 \theta \) for \( 0 < \theta < \pi \), we can use the Arithmetic Mean-Geometric Mean (AM-GM) inequality.

### Step-by-Step Solution:

1. **Identify the terms**: We have two terms, \( \sin^5 \theta \) and \( \csc^5 \theta \).

2. **Apply the AM-GM Inequality**: According to the AM-GM inequality, for any non-negative real numbers \( a \) and \( b \):
\[ \frac{a + b}{2} \geq \sqrt{ab} \]
Here, let \( a = \sin^5 \theta \) and \( b = \csc^5 \theta \).

3. **Set up the inequality**:
\[ \frac{\sin^5 \theta + \csc^5 \theta}{2} \geq \sqrt{\sin^5 \theta \cdot \csc^5 \theta} \]

4. **Simplify the right-hand side**: We know that \( \csc \theta = \frac{1}{\sin \theta} \), so:
\[ \csc^5 \theta = \left(\frac{1}{\sin \theta}\right)^5 = \frac{1}{\sin^5 \theta} \]
Therefore:
\[ \sin^5 \theta \cdot \csc^5 \theta = \sin^5 \theta \cdot \frac{1}{\sin^5 \theta} = 1 \]

5. **Substitute back into the inequality**:
\[ \frac{\sin^5 \theta + \csc^5 \theta}{2} \geq \sqrt{1} = 1 \]

6. **Multiply both sides by 2**:
\[ \sin^5 \theta + \csc^5 \theta \geq 2 \]

7. **Conclusion**: The minimum value of \( \sin^5 \theta + \csc^5 \theta \) is \( 2 \).

### Final Answer:
The minimum value of \( \sin^5 \theta + \csc^5 \theta \) is \( 2 \).