To solve the trigonometric equation \(\sin(30^\circ + A) - \sin(60^\circ - A) = \sqrt{3}\sin(A)\), we can use the sum-to-product identities. Let's proceed with the solution.

### Step 1: Apply the sum-to-product identities
The sum-to-product identity for the difference of sines is:
\[
\sin x - \sin y = 2 \cos \left(\frac{x + y}{2}\right) \sin \left(\frac{x - y}{2}\right)
\]
Here, \( x = 30^\circ + A \) and \( y = 60^\circ - A \).

Applying the identity:
\[
\sin(30^\circ + A) - \sin(60^\circ - A) = 2 \cos \left(\frac{(30^\circ + A) + (60^\circ - A)}{2}\right) \sin \left(\frac{(30^\circ + A) - (60^\circ - A)}{2}\right)
\]

### Step 2: Simplify the expressions
Simplify the arguments of the cosine and sine functions:

\[
\cos \left(\frac{(30^\circ + A) + (60^\circ - A)}{2}\right) = \cos \left(\frac{90^\circ}{2}\right) = \cos(45^\circ) = \frac{\sqrt{2}}{2}
\]

\[
\sin \left(\frac{(30^\circ + A) - (60^\circ - A)}{2}\right) = \sin \left(\frac{-30^\circ + 2A}{2}\right) = \sin \left(-15^\circ + A\right)
\]

Thus:
\[
\sin(30^\circ + A) - \sin(60^\circ - A) = 2 \cdot \frac{\sqrt{2}}{2} \cdot \sin(-15^\circ + A) = \sqrt{2} \sin(-15^\circ + A)
\]

### Step 3: Set the equation equal to \(\sqrt{3}\sin(A)\)
Now, substitute this back into the original equation:

\[
\sqrt{2} \sin(-15^\circ + A) = \sqrt{3}\sin(A)
\]

### Step 4: Solve the equation
This equation implies that for both sides to be equal, either the angles or the sine functions themselves should have some relationship. Solving this directly requires a more detailed analysis or specific techniques, but generally, one would look for values of \(A\) that satisfy this equation.

You can explore specific values of \(A\) or solve it graphically or numerically, depending on the desired approach.

Would you like further details, or do you have any questions about the steps?

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Here are some related questions you might explore:
1. What are sum-to-product identities, and how are they derived?
2. How would you solve \(\sin x - \sin y = k\) for different constants \(k\)?
3. What happens if we change the angles in the original equation?
4. Can we solve the equation using a different trigonometric identity?
5. How can we verify the solutions graphically?

**Tip:** When working with trigonometric identities, it helps to familiarize yourself with sum-to-product, product-to-sum, and other transformations, as they are useful in simplifying and solving complex equations.

Learn how to solve the trigonometric equation sin(30+A) - sin(60-A) = sqrt(3)sin(A) step-by-step. This problem involves advanced trigonometric identities and equations. Suitable for advanced high school students and those studying trigonometry.

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