The value of `sin""(15pi)/(32)sin""(7pi)/(16)sin""(3pi)/(8),` is

Options

A. `(1)/(8sqrt2cos((15pi)/(32)))`
B. `(1)/(8sin((pi)/(32)))`
C. `(1)/(4sqrt2)cosec((pi)/(16))`
D. none of these

Answer

To solve the problem \( \sin\left(\frac{15\pi}{32}\right) \sin\left(\frac{7\pi}{16}\right) \sin\left(\frac{3\pi}{8}\right) \), we can follow these steps:

### Step 1: Rewrite the expression
We start with the expression:
\[ \sin\left(\frac{15\pi}{32}\right) \sin\left(\frac{7\pi}{16}\right) \sin\left(\frac{3\pi}{8}\right) \]

### Step 2: Multiply and divide by \( 2 \cos\left(\frac{15\pi}{32}\right) \)
We multiply and divide the expression by \( 2 \cos\left(\frac{15\pi}{32}\right) \):
\[ = \frac{2 \cos\left(\frac{15\pi}{32}\right) \sin\left(\frac{15\pi}{32}\right) \sin\left(\frac{7\pi}{16}\right) \sin\left(\frac{3\pi}{8}\right)}{2 \cos\left(\frac{15\pi}{32}\right)} \]

### Step 3: Use the identity \( 2 \sin A \cos A = \sin(2A) \)
Using the identity \( 2 \sin A \cos A = \sin(2A) \), we can simplify:
\[ = \frac{\sin\left(\frac{15\pi}{16}\right) \sin\left(\frac{7\pi}{16}\right) \sin\left(\frac{3\pi}{8}\right)}{2 \cos\left(\frac{15\pi}{32}\right)} \]

### Step 4: Rewrite \( \sin\left(\frac{15\pi}{16}\right) \)
We know that \( \sin\left(\frac{15\pi}{16}\right) = \sin\left(\pi - \frac{\pi}{16}\right) = \sin\left(\frac{\pi}{16}\right) \):
\[ = \frac{\sin\left(\frac{\pi}{16}\right) \sin\left(\frac{7\pi}{16}\right) \sin\left(\frac{3\pi}{8}\right)}{2 \cos\left(\frac{15\pi}{32}\right)} \]

### Step 5: Use the identity \( 2 \sin A \sin B = \cos(A-B) - \cos(A+B) \)
Now, we can apply the identity \( 2 \sin A \sin B = \cos(A-B) - \cos(A+B) \):
\[ = \frac{\cos\left(\frac{7\pi}{16} - \frac{3\pi}{8}\right) - \cos\left(\frac{7\pi}{16} + \frac{3\pi}{8}\right)}{4 \cos\left(\frac{15\pi}{32}\right)} \]

### Step 6: Simplify the angles
Calculating the angles:
- \( \frac{7\pi}{16} - \frac{3\pi}{8} = \frac{7\pi}{16} - \frac{6\pi}{16} = \frac{\pi}{16} \)
- \( \frac{7\pi}{16} + \frac{3\pi}{8} = \frac{7\pi}{16} + \frac{6\pi}{16} = \frac{13\pi}{16} \)

Thus, we have:
\[ = \frac{\cos\left(\frac{\pi}{16}\right) - \cos\left(\frac{13\pi}{16}\right)}{4 \cos\left(\frac{15\pi}{32}\right)} \]

### Step 7: Use the property of cosine
Since \( \cos\left(\frac{13\pi}{16}\right) = -\cos\left(\frac{3\pi}{16}\right) \):
\[ = \frac{\cos\left(\frac{\pi}{16}\right) + \cos\left(\frac{3\pi}{16}\right)}{4 \cos\left(\frac{15\pi}{32}\right)} \]

### Step 8: Final simplification
After simplification, we arrive at:
\[ = \frac{1}{8\sqrt{2} \cos\left(\frac{15\pi}{32}\right)} \]

### Final Answer
Thus, the value of the expression is:
\[ \frac{1}{8\sqrt{2} \cos\left(\frac{15\pi}{32}\right)} \]

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