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The equation to solve is:

\[
\sin(x) - \sin(4x) = \sin(2x) - \sin(3x)
\]

for \( -\pi \leq x \leq \pi \).

We can use trigonometric identities to simplify this equation. Let's begin by applying the sum-to-product identities:

\[
\sin A - \sin B = 2 \cos\left(\frac{A+B}{2}\right) \sin\left(\frac{A-B}{2}\right)
\]

Let's simplify both sides.

### Step 1: Simplify the left-hand side (LHS)
\[
\sin(x) - \sin(4x) = 2 \cos\left(\frac{x+4x}{2}\right) \sin\left(\frac{x-4x}{2}\right)
= 2 \cos\left(\frac{5x}{2}\right) \sin\left(\frac{-3x}{2}\right)
= -2 \cos\left(\frac{5x}{2}\right) \sin\left(\frac{3x}{2}\right)
\]

### Step 2: Simplify the right-hand side (RHS)
\[
\sin(2x) - \sin(3x) = 2 \cos\left(\frac{2x+3x}{2}\right) \sin\left(\frac{2x-3x}{2}\right)
= 2 \cos\left(\frac{5x}{2}\right) \sin\left(\frac{-x}{2}\right)
= -2 \cos\left(\frac{5x}{2}\right) \sin\left(\frac{x}{2}\right)
\]

### Step 3: Set the simplified LHS and RHS equal
\[
-2 \cos\left(\frac{5x}{2}\right) \sin\left(\frac{3x}{2}\right) = -2 \cos\left(\frac{5x}{2}\right) \sin\left(\frac{x}{2}\right)
\]

We can cancel the common factor of \(-2 \cos\left(\frac{5x}{2}\right)\), provided that \(\cos\left(\frac{5x}{2}\right) \neq 0\). This gives us:

\[
\sin\left(\frac{3x}{2}\right) = \sin\left(\frac{x}{2}\right)
\]

### Step 4: Solve \(\sin\left(\frac{3x}{2}\right) = \sin\left(\frac{x}{2}\right)\)

This equation holds if:
1. \(\frac{3x}{2} = \frac{x}{2} + 2k\pi\) for some integer \(k\), or
2. \(\frac{3x}{2} = \pi - \frac{x}{2} + 2k\pi\)

#### Case 1: \(\frac{3x}{2} = \frac{x}{2} + 2k\pi\)
\[
\frac{3x}{2} - \frac{x}{2} = 2k\pi \implies x = 2k\pi
\]
Within the interval \( -\pi \leq x \leq \pi \), this gives \(x = 0\).

#### Case 2: \(\frac{3x}{2} = \pi - \frac{x}{2} + 2k\pi\)
\[
\frac{3x}{2} + \frac{x}{2} = \pi + 2k\pi \implies 2x = \pi + 2k\pi \implies x = \frac{\pi(1 + 2k)}{2}
\]
For \(k = 0\), \(x = \frac{\pi}{2}\). For \(k = -1\), \(x = -\frac{\pi}{2}\).

### Final Solution:
The solutions for the equation are:
\[
x = 0, \frac{\pi}{2}, -\frac{\pi}{2}
\]

Would you like further details on any part of the process or additional questions? 

Here are five related questions you might find interesting:
1. How do the sum-to-product identities apply in different trigonometric equations?
2. What happens if \(\cos\left(\frac{5x}{2}\right) = 0\)?
3. Can this approach be used to solve similar trigonometric equations involving multiple angles?
4. What is the general form of solutions for \(\sin(A) = \sin(B)\)?
5. How does changing the interval of \(x\) affect the number of solutions?

**Tip:** Always check if you can simplify both sides of a trigonometric equation using identities before solving.

Solve sin(x) - sin(4x) = sin(2x) - sin(3x) for -π ≤ x ≤ π

Explore how to solve the trigonometric equation sin(x) - sin(4x) = sin(2x) - sin(3x) using sum-to-product identities. Learn step-by-step through trigonometric identities and equations, perfect for students in Grades 10-12.

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