sin x minus sin 3x upon sin x square - cos x square is equivalent to

We are given the expression:

\[
\frac{\sin x - \sin 3x}{\sin^2 x - \cos^2 x}
\]

### Step 1: Simplify the numerator
The numerator, \(\sin x - \sin 3x\), can be simplified using the identity for the difference of sines:

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

Let \(A = 3x\) and \(B = x\):

\[
\sin x - \sin 3x = 2 \cos\left(\frac{3x + x}{2}\right) \sin\left(\frac{x - 3x}{2}\right)
\]
\[
= 2 \cos(2x) \sin(-x)
\]
Since \(\sin(-x) = -\sin(x)\), this becomes:

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

### Step 2: Simplify the denominator
The denominator, \(\sin^2 x - \cos^2 x\), is a known identity:

\[
\sin^2 x - \cos^2 x = -\cos(2x)
\]

### Step 3: Substitute the simplified expressions
Now substitute the simplified numerator and denominator back into the expression:

\[
\frac{-2 \cos(2x) \sin(x)}{-\cos(2x)}
\]

### Step 4: Simplify the fraction
The \(-\cos(2x)\) terms cancel out:

\[
= 2 \sin(x)
\]

Thus, the given expression simplifies to:

\[
2 \sin(x)
\]

### Final Answer:
\[
\frac{\sin x - \sin 3x}{\sin^2 x - \cos^2 x} = 2 \sin(x)
\]

Would you like further details or have any questions?

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Here are 5 related questions to explore this concept further:

1. What are some other trigonometric identities involving \(\sin\) and \(\cos\)?
2. How does the identity \(\sin A - \sin B\) help in simplifying expressions?
3. What is the geometric interpretation of \(\sin^2 x - \cos^2 x\)?
4. Can you derive similar results using different trigonometric identities?
5. How would you apply this simplification in solving integrals?

**Tip:** Remember, simplifying trigonometric expressions often involves applying fundamental identities like sum/difference formulas and Pythagorean identities!

This guide walks through the process of simplifying the trigonometric expression sin x - sin 3x over sin^2 x - cos^2 x using difference of sines and Pythagorean identities. It provides a step-by-step solution suitable for high school students in Grades 10-12.

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