To simplify the expression \( k = 1 \prod_{n-1} \sin\left(\frac{2\pi n}{k}\right) \), let's carefully rewrite it and address the product notation.

1. The expression involves the product notation: \( \prod_{n-1} \), which implies taking the product of terms for values of \( n \) in a given range.
   
2. You have \( \sin\left(\frac{2\pi n}{k}\right) \), which is a trigonometric sine function, applied to an angle expressed as a fraction of \( 2\pi \).

### Key questions for simplification:
- What is the exact range of \( n \)? (e.g., is it from 1 to \( k-1 \)?)
- Is \( k \) a constant in this expression?
- Are there any assumptions about \( n \), such as whether it represents an integer or another set of values?

### Here's a typical simplification step:
If \( n \) takes integer values from 1 to \( k-1 \), the product \( \prod_{n=1}^{k-1} \sin\left(\frac{2\pi n}{k}\right) \) is a known result in trigonometric identities and can be simplified using a known product formula:

\[
\prod_{n=1}^{k-1} \sin\left(\frac{2\pi n}{k}\right) = \frac{k}{2^{k-1}}
\]

Thus, the simplified version of the expression would be:

\[
\prod_{n=1}^{k-1} \sin\left(\frac{2\pi n}{k}\right) = \frac{k}{2^{k-1}}
\]

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Would you like further details or have any questions on how this simplification works?

Here are 5 related questions:
1. How does the product identity for sine functions work in this context?
2. Can we use this identity for non-integer values of \( k \)?
3. What happens if \( n \) has a different range or if \( k \) is modified?
4. How does this result relate to trigonometric product formulas in general?
5. What role does the period of the sine function play in simplifying such products?

**Tip:** The product of sine terms often simplifies using known formulas related to trigonometric identities, particularly when the arguments are symmetrically spaced around the unit circle.

Explore the simplification of the product of sine functions in the expression k=1∏(n-1) sin(2πn/k). This detailed guide covers trigonometric identities, product notation, and simplification techniques, suitable for students in Grades 11-12.

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