If `sin theta + cosec theta =2`,then the value of `sin^5theta cosec^5 theta`,when `0^@ le theta le 90^@`, is

Options

A. 0
B. 1
C. 10
D. 2

Answer

To solve the problem, we need to find the value of \( \sin^5 \theta \cdot \csc^5 \theta \) given that \( \sin \theta + \csc \theta = 2 \) and \( 0^\circ \leq \theta \leq 90^\circ \).

### Step-by-Step Solution:

1. **Understanding the equation**:
We start with the equation:
\[ \sin \theta + \csc \theta = 2 \]
Recall that \( \csc \theta = \frac{1}{\sin \theta} \). Therefore, we can rewrite the equation as:
\[ \sin \theta + \frac{1}{\sin \theta} = 2 \]

2. **Letting \( x = \sin \theta \)**:
Substitute \( x \) for \( \sin \theta \):
\[ x + \frac{1}{x} = 2 \]

3. **Multiplying through by \( x \)**:
Multiply both sides by \( x \) (noting that \( x \) cannot be zero since \( \sin \theta \) is positive in the given range):
\[ x^2 + 1 = 2x \]

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

5. **Factoring the quadratic**:
This can be factored as:
\[ (x - 1)^2 = 0 \]
Thus, we find:
\[ x - 1 = 0 \implies x = 1 \]

6. **Finding \( \sin \theta \)**:
Since \( x = \sin \theta \), we have:
\[ \sin \theta = 1 \]
This occurs when:
\[ \theta = 90^\circ \]

7. **Calculating \( \sin^5 \theta \cdot \csc^5 \theta \)**:
Now we need to find:
\[ \sin^5 \theta \cdot \csc^5 \theta \]
Since \( \sin \theta = 1 \):
\[ \csc \theta = \frac{1}{\sin \theta} = 1 \]
Therefore:
\[ \sin^5 \theta = 1^5 = 1 \]
and
\[ \csc^5 \theta = 1^5 = 1 \]
Thus:
\[ \sin^5 \theta \cdot \csc^5 \theta = 1 \cdot 1 = 1 \]

### Final Answer:
The value of \( \sin^5 \theta \cdot \csc^5 \theta \) is \( 1 \).