The value of `sin""(pi)/(10)sin""(13pi)/(10)` is

Options

A. `(1)/(2)`
B. `-(1)/(2)`
C. `-(1)/(4)`
D. 1

Explanation

To find the value of \( \sin\left(\frac{\pi}{10}\right) \sin\left(\frac{13\pi}{10}\right) \), we can follow these steps:

### Step 1: Rewrite \( \sin\left(\frac{13\pi}{10}\right) \)
We can express \( \sin\left(\frac{13\pi}{10}\right) \) in terms of a known angle:
\[ \sin\left(\frac{13\pi}{10}\right) = \sin\left(\pi + \frac{3\pi}{10}\right) \]
Using the identity \( \sin(\pi + \theta) = -\sin(\theta) \), we have:
\[ \sin\left(\frac{13\pi}{10}\right) = -\sin\left(\frac{3\pi}{10}\right) \]

### Step 2: Substitute back into the expression
Now, substitute this back into the original expression:
\[ \sin\left(\frac{\pi}{10}\right) \sin\left(\frac{13\pi}{10}\right) = \sin\left(\frac{\pi}{10}\right) \cdot \left(-\sin\left(\frac{3\pi}{10}\right)\right) \]
This simplifies to:
\[ -\sin\left(\frac{\pi}{10}\right) \sin\left(\frac{3\pi}{10}\right) \]

### Step 3: Use the identity for \( \sin A \sin B \)
We can use the product-to-sum identities:
\[ \sin A \sin B = \frac{1}{2} \left( \cos(A - B) - \cos(A + B) \right) \]
Let \( A = \frac{\pi}{10} \) and \( B = \frac{3\pi}{10} \):
\[ -\sin\left(\frac{\pi}{10}\right) \sin\left(\frac{3\pi}{10}\right) = -\frac{1}{2} \left( \cos\left(\frac{\pi}{10} - \frac{3\pi}{10}\right) - \cos\left(\frac{\pi}{10} + \frac{3\pi}{10}\right) \right) \]

### Step 4: Calculate the angles
Calculating the angles:
\[ \frac{\pi}{10} - \frac{3\pi}{10} = -\frac{2\pi}{10} = -\frac{\pi}{5} \]
\[ \frac{\pi}{10} + \frac{3\pi}{10} = \frac{4\pi}{10} = \frac{2\pi}{5} \]

### Step 5: Substitute the angles back
Now substituting these angles back into the expression:
\[ -\frac{1}{2} \left( \cos\left(-\frac{\pi}{5}\right) - \cos\left(\frac{2\pi}{5}\right) \right) \]
Using the fact that \( \cos(-x) = \cos(x) \):
\[ -\frac{1}{2} \left( \cos\left(\frac{\pi}{5}\right) - \cos\left(\frac{2\pi}{5}\right) \right) \]

### Step 6: Use known values of cosines
We know that:
\[ \cos\left(\frac{\pi}{5}\right) = \frac{1 + \sqrt{5}}{4}, \quad \cos\left(\frac{2\pi}{5}\right) = \frac{\sqrt{5} - 1}{4} \]
Substituting these values:
\[ -\frac{1}{2} \left( \frac{1 + \sqrt{5}}{4} - \frac{\sqrt{5} - 1}{4} \right) \]
Simplifying this:
\[ -\frac{1}{2} \left( \frac{1 + \sqrt{5} - \sqrt{5} + 1}{4} \right) = -\frac{1}{2} \left( \frac{2}{4} \right) = -\frac{1}{2} \cdot \frac{1}{2} = -\frac{1}{4} \]

### Final Answer
Thus, the value of \( \sin\left(\frac{\pi}{10}\right) \sin\left(\frac{13\pi}{10}\right) \) is:
\[ \boxed{-\frac{1}{4}} \]