The value of `sin(pi+theta)sin(pi-theta)cosec^(2)theta` is equal to

Options

A. `-1`
B. 0
C. `sin theta`
D. none of these

Answer

To solve the problem, we need to evaluate the expression \( \sin(\pi + \theta) \sin(\pi - \theta) \csc^2(\theta) \).

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

### Step 2: Evaluate \( \sin(\pi - \theta) \)
Using the sine subtraction formula, we have:
\[ \sin(\pi - \theta) = \sin(\theta) \]

### Step 3: Substitute the values into the expression
Now, substituting the values from Steps 1 and 2 into the expression:
\[ \sin(\pi + \theta) \sin(\pi - \theta) \csc^2(\theta) = (-\sin(\theta)) \sin(\theta) \csc^2(\theta) \]

### Step 4: Simplify the expression
Now, we simplify the expression:
\[ = -\sin^2(\theta) \csc^2(\theta) \]
Since \( \csc(\theta) = \frac{1}{\sin(\theta)} \), we have:
\[ \csc^2(\theta) = \frac{1}{\sin^2(\theta)} \]
Thus, substituting this in gives:
\[ = -\sin^2(\theta) \cdot \frac{1}{\sin^2(\theta)} = -1 \]

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