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The image shows a trigonometric identity that needs to be proven. The equation is:

\[
\sin(A + 60^\circ) \sin(A - 60^\circ) = \frac{\sin(3A)}{4 \sin(A)}
\]

### Let's go through the steps to prove this:

1. **Expand the left-hand side** using the product-to-sum identities:
   \[
   \sin(A + 60^\circ) \sin(A - 60^\circ) = \frac{1}{2} [\cos(60^\circ - A) - \cos(60^\circ + A)]
   \]

2. Simplify using the cosine addition and subtraction formulas:
   \[
   \cos(60^\circ - A) = \cos(60^\circ) \cos(A) + \sin(60^\circ) \sin(A)
   \]
   \[
   \cos(60^\circ + A) = \cos(60^\circ) \cos(A) - \sin(60^\circ) \sin(A)
   \]

   Since \(\cos(60^\circ) = \frac{1}{2}\) and \(\sin(60^\circ) = \frac{\sqrt{3}}{2}\), substitute these values:
   \[
   \sin(A + 60^\circ) \sin(A - 60^\circ) = \frac{1}{2} \left[ \frac{1}{2} \cos(A) + \frac{\sqrt{3}}{2} \sin(A) - \left( \frac{1}{2} \cos(A) - \frac{\sqrt{3}}{2} \sin(A) \right) \right]
   \]

3. Simplify the expression:
   \[
   \sin(A + 60^\circ) \sin(A - 60^\circ) = \frac{1}{2} \left[ \sqrt{3} \sin(A) \right] = \frac{\sqrt{3}}{2} \sin(A)
   \]

4. **Compare with the right-hand side**:
   Now, let's examine the right-hand side \(\frac{\sin(3A)}{4 \sin(A)}\). Using the triple angle identity for sine:
   \[
   \sin(3A) = 3 \sin(A) - 4 \sin^3(A)
   \]

   Thus, the right-hand side becomes:
   \[
   \frac{3 \sin(A) - 4 \sin^3(A)}{4 \sin(A)}
   \]

   Simplifying:
   \[
   \frac{\sin(3A)}{4 \sin(A)} = \frac{3 - 4 \sin^2(A)}{4}
   \]

To fully equate both sides, we would check specific values of \(A\) or perform further simplifications to conclude the proof.

Do you need further details on any of these steps? Let me know!

### Related Questions:
1. What is the product-to-sum identity and how is it derived?
2. How do you prove the triple angle identity for sine?
3. Can this identity be extended to different angle measures?
4. What other trigonometric identities involve angle addition and subtraction?
5. How does the cosine identity differ when adding or subtracting angles?

**Tip:** Remember to familiarize yourself with the key trigonometric identities like sum-to-product, product-to-sum, and triple angle identities for faster problem-solving.

Show that sin(A + 60°) sin(A - 60°) = sin(3A) / (4 sin(A))

Learn how to prove the trigonometric identity sin(A + 60°) sin(A - 60°) = sin(3A) / (4 sin(A)) using product-to-sum and triple angle identities. This problem covers key trigonometric concepts, including angle addition and the triple angle formula for sine. Suitable for students in Grades 11-12.

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