Statement:1 : Minimum value of `sinx + " cosec "x` is 2 for all `x in (0,pi)` and Statement 2: Min`(sinx + " cosec "x)=2, AA x in R`

Options

A. Statement-1 is true, Statement-2 is true, Statement-2 is a correct explanation for statement-1
B. Statement-1 is true, Statement-2 is true, Statement-2 is not a correct expanation for statement-1
C. Statement-1 is True, Statement-2 is false
D. Statement-1 is False, Statement-2 is true

Answer

To solve the problem, we need to analyze the two statements regarding the function \( f(x) = \sin x + \csc x \).

### Step 1: Understanding the Function
We start with the function:
\[ f(x) = \sin x + \csc x \]
where \( \csc x = \frac{1}{\sin x} \).

### Step 2: Domain of the Function
The function \( f(x) \) is defined for \( x \) in the interval \( (0, \pi) \) because \( \sin x \) is positive in this interval, making \( \csc x \) defined.

### Step 3: Applying AM-GM Inequality
We apply the Arithmetic Mean-Geometric Mean (AM-GM) inequality:
\[ \frac{a + b}{2} \geq \sqrt{ab} \]
for \( a = \sin x \) and \( b = \csc x \). Thus,
\[ \frac{\sin x + \csc x}{2} \geq \sqrt{\sin x \cdot \csc x} \]
Since \( \csc x = \frac{1}{\sin x} \), we have:
\[ \sin x \cdot \csc x = 1 \]
This leads to:
\[ \frac{\sin x + \csc x}{2} \geq \sqrt{1} = 1 \]
Multiplying both sides by 2 gives:
\[ \sin x + \csc x \geq 2 \]

### Step 4: Finding the Minimum Value
The equality in the AM-GM inequality holds when \( a = b \), which means:
\[ \sin x = \csc x \implies \sin^2 x = 1 \implies \sin x = 1 \]
This occurs at \( x = \frac{\pi}{2} \).

### Step 5: Evaluating the Function at \( x = \frac{\pi}{2} \)
Calculating \( f\left(\frac{\pi}{2}\right) \):
\[ f\left(\frac{\pi}{2}\right) = \sin\left(\frac{\pi}{2}\right) + \csc\left(\frac{\pi}{2}\right) = 1 + 1 = 2 \]

### Conclusion
Thus, the minimum value of \( f(x) = \sin x + \csc x \) for \( x \in (0, \pi) \) is indeed 2, occurring at \( x = \frac{\pi}{2} \).

### Final Statements
- **Statement 1**: Minimum value of \( \sin x + \csc x \) is 2 for all \( x \in (0, \pi) \) is **true**.
- **Statement 2**: Minimum of \( \sin x + \csc x = 2 \) for all \( x \in \mathbb{R} \) is **false** because the function is not defined for \( x \in \mathbb{R} \) (only defined for \( x \in (0, \pi) \)).