The value of `sin(pi+x).sin(pi-x)."cosec"^(2)x` is

Options

A. 0
B. 1
C. `-1`
D. 2

Answer

To solve the expression \( \sin(\pi + x) \cdot \sin(\pi - x) \cdot \csc^2(x) \), we will follow these steps:

### Step 1: Evaluate \( \sin(\pi + x) \)
Using the sine addition formula:
\[ \sin(\pi + x) = -\sin(x) \]

### Step 2: Evaluate \( \sin(\pi - x) \)
Using the sine subtraction formula:
\[ \sin(\pi - x) = \sin(x) \]

### Step 3: Substitute the values into the expression
Now we substitute the values from Steps 1 and 2 into the original expression:
\[ \sin(\pi + x) \cdot \sin(\pi - x) \cdot \csc^2(x) = (-\sin(x)) \cdot \sin(x) \cdot \csc^2(x) \]

### Step 4: Simplify the expression
We know that:
\[ \csc(x) = \frac{1}{\sin(x)} \implies \csc^2(x) = \frac{1}{\sin^2(x)} \]
Thus, we can rewrite the expression:
\[ (-\sin(x)) \cdot \sin(x) \cdot \csc^2(x) = -\sin(x) \cdot \sin(x) \cdot \frac{1}{\sin^2(x)} \]

### Step 5: Cancel out \( \sin^2(x) \)
The \( \sin^2(x) \) terms cancel out:
\[ -\sin^2(x) \cdot \frac{1}{\sin^2(x)} = -1 \]

### Final Answer
Thus, the value of the expression \( \sin(\pi + x) \cdot \sin(\pi - x) \cdot \csc^2(x) \) is:
\[ \boxed{-1} \]

---