The value of `sin(pi/14)sin(3pi/14)sin(5pi/14)sin(7pi/14)sin(9pi/14)sin(11pi/14)sin(13pi/14)` is equal to___________

Explanation

To find the value of the expression

\[ \sin\left(\frac{\pi}{14}\right) \sin\left(\frac{3\pi}{14}\right) \sin\left(\frac{5\pi}{14}\right) \sin\left(\frac{7\pi}{14}\right) \sin\left(\frac{9\pi}{14}\right) \sin\left(\frac{11\pi}{14}\right) \sin\left(\frac{13\pi}{14}\right), \]

we can simplify the expression step by step.

### Step 1: Simplify \(\sin\left(\frac{7\pi}{14}\right)\)
First, we note that

\[ \sin\left(\frac{7\pi}{14}\right) = \sin\left(\frac{\pi}{2}\right) = 1. \]

So we can rewrite the expression as:

\[ \sin\left(\frac{\pi}{14}\right) \sin\left(\frac{3\pi}{14}\right) \sin\left(\frac{5\pi}{14}\right) \cdot 1 \cdot \sin\left(\frac{9\pi}{14}\right) \sin\left(\frac{11\pi}{14}\right) \sin\left(\frac{13\pi}{14}\right). \]

### Step 2: Use the property of sine
Next, we use the property of sine that states:

\[ \sin\left(\pi - x\right) = \sin x. \]

This allows us to rewrite:

- \(\sin\left(\frac{9\pi}{14}\right) = \sin\left(\pi - \frac{5\pi}{14}\right) = \sin\left(\frac{5\pi}{14}\right)\),
- \(\sin\left(\frac{11\pi}{14}\right) = \sin\left(\pi - \frac{3\pi}{14}\right) = \sin\left(\frac{3\pi}{14}\right)\),
- \(\sin\left(\frac{13\pi}{14}\right) = \sin\left(\pi - \frac{\pi}{14}\right) = \sin\left(\frac{\pi}{14}\right)\).

### Step 3: Substitute back into the expression
Now substituting these back into the expression, we have:

\[ \sin\left(\frac{\pi}{14}\right) \sin\left(\frac{3\pi}{14}\right) \sin\left(\frac{5\pi}{14}\right) \cdot 1 \cdot \sin\left(\frac{5\pi}{14}\right) \sin\left(\frac{3\pi}{14}\right) \sin\left(\frac{\pi}{14}\right). \]

### Step 4: Combine like terms
This simplifies to:

\[ \left(\sin\left(\frac{\pi}{14}\right)\right)^2 \left(\sin\left(\frac{3\pi}{14}\right)\right)^2 \left(\sin\left(\frac{5\pi}{14}\right)\right)^2. \]

### Step 5: Use a known product identity
There is a known identity for the product of sines:

\[ \sin\left(\frac{\pi}{14}\right) \sin\left(\frac{3\pi}{14}\right) \sin\left(\frac{5\pi}{14}\right) = \frac{\sqrt{7}}{8}. \]

### Step 6: Square the result
Thus, we have:

\[ \left(\sin\left(\frac{\pi}{14}\right) \sin\left(\frac{3\pi}{14}\right) \sin\left(\frac{5\pi}{14}\right)\right)^2 = \left(\frac{\sqrt{7}}{8}\right)^2 = \frac{7}{64}. \]

### Final Answer
Therefore, the value of

\[ \sin\left(\frac{\pi}{14}\right) \sin\left(\frac{3\pi}{14}\right) \sin\left(\frac{5\pi}{14}\right) \sin\left(\frac{7\pi}{14}\right) \sin\left(\frac{9\pi}{14}\right) \sin\left(\frac{11\pi}{14}\right) \sin\left(\frac{13\pi}{14}\right) = \frac{7}{64}. \]