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To find an equivalent expression for \( 2 \sin \left( \frac{5\pi}{12} \right) \cos \left( \frac{5\pi}{12} \right) \) using a double-angle formula, we can use the identity:

\[
\sin(2\theta) = 2 \sin(\theta) \cos(\theta)
\]

In this case, let \( \theta = \frac{5\pi}{12} \). Then:

\[
2 \sin \left( \frac{5\pi}{12} \right) \cos \left( \frac{5\pi}{12} \right) = \sin \left( 2 \cdot \frac{5\pi}{12} \right)
\]

Simplifying \( 2 \cdot \frac{5\pi}{12} \):

\[
2 \cdot \frac{5\pi}{12} = \frac{5\pi}{6}
\]

Thus:

\[
2 \sin \left( \frac{5\pi}{12} \right) \cos \left( \frac{5\pi}{12} \right) = \sin \left( \frac{5\pi}{6} \right)
\]

The correct answer is therefore:

- **\( \sin \left( \frac{5\pi}{6} \right) \)**

Would you like more details on the double-angle formula, or have any further questions?

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### Related Questions

1. What is the double-angle formula for cosine?
2. How do you simplify trigonometric expressions with half-angles?
3. How can the double-angle formula be used to solve trigonometric equations?
4. What are some applications of trigonometric identities in real-world problems?
5. How do we use sum-to-product identities in trigonometry?

### Tip
Always verify if a trigonometric identity can simplify the expression before calculating it directly; it can save time and reduce error.

Use a double-angle formula to state an equivalent expression. 2 sin(5π/12) cos(5π/12).

Explore how to use the double-angle formula to find an equivalent expression for 2 sin(5π/12) cos(5π/12). Learn the steps and key trigonometric identities involved in this advanced high school problem, ideal for students in Grades 11-12.

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