If `0ltthetaltpi` , then the minmum value of `sin^(3) theta+cosec^(3) theta +2` is

Options

A. 0
B. 2
C. 4
D. 8

Answer

To find the minimum value of the expression \( \sin^3 \theta + \csc^3 \theta + 2 \) for \( 0 < \theta < \pi \), we can follow these steps:

### Step 1: Rewrite the Expression
Let \( y = \sin^3 \theta + \csc^3 \theta + 2 \).

### Step 2: Substitute \( x \)
Define \( x = \sin^3 \theta + \csc^3 \theta \). Thus, we can rewrite the expression as:
\[ y = x + 2 \]

### Step 3: Analyze \( x \)
Since \( \sin \theta \) is positive in the interval \( 0 < \theta < \pi \), both \( \sin^3 \theta \) and \( \csc^3 \theta \) are also positive. Therefore, we can apply the Arithmetic Mean-Geometric Mean (AM-GM) inequality:
\[ \frac{\sin^3 \theta + \csc^3 \theta}{2} \geq \sqrt{\sin^3 \theta \cdot \csc^3 \theta} \]

### Step 4: Simplify the AM-GM Inequality
The product \( \sin^3 \theta \cdot \csc^3 \theta = \sin^3 \theta \cdot \frac{1}{\sin^3 \theta} = 1 \). Thus, we have:
\[ \frac{\sin^3 \theta + \csc^3 \theta}{2} \geq \sqrt{1} = 1 \]
This implies:
\[ \sin^3 \theta + \csc^3 \theta \geq 2 \]

### Step 5: Find Minimum Value of \( x \)
From the previous step, we find that the minimum value of \( x \) is:
\[ x \geq 2 \]

### Step 6: Substitute Back to Find Minimum \( y \)
Now substituting back into the expression for \( y \):
\[ y = x + 2 \]
Thus, the minimum value of \( y \) is:
\[ y \geq 2 + 2 = 4 \]

### Conclusion
The minimum value of \( \sin^3 \theta + \csc^3 \theta + 2 \) is \( 4 \).

### Final Answer
The minimum value is \( \boxed{4} \).

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