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The given expression is:

\[
\frac{\sin^4(x) - \cos^4(x)}{\sin^2(x) - \cos^2(x)}
\]

To simplify this, we can recognize that the numerator is a difference of squares. The difference of squares formula states:

\[
a^2 - b^2 = (a - b)(a + b)
\]

Here, we treat \(\sin^4(x)\) and \(\cos^4(x)\) as \((\sin^2(x))^2\) and \((\cos^2(x))^2\). Therefore, we can apply the difference of squares formula to the numerator:

\[
\sin^4(x) - \cos^4(x) = (\sin^2(x) - \cos^2(x))(\sin^2(x) + \cos^2(x))
\]

Next, we can simplify the expression:

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

The term \(\sin^2(x) - \cos^2(x)\) appears in both the numerator and the denominator, so we can cancel it out:

\[
\sin^2(x) + \cos^2(x)
\]

Finally, we know from the Pythagorean identity that:

\[
\sin^2(x) + \cos^2(x) = 1
\]

Thus, the simplified expression is:

\[
1
\]

Let me know if you would like further details or have any questions. Here are 5 related questions to extend this concept:

1. How does the difference of squares formula apply to trigonometric identities?
2. Can you apply similar simplifications to higher-order trigonometric powers?
3. What are some common trigonometric identities used in calculus?
4. How would you simplify \(\sin^6(x) - \cos^6(x)\)?
5. What is the importance of the Pythagorean identity in trigonometry?

**Tip:** Always look for identities or factoring opportunities to simplify complex trigonometric expressions.

Simplify the expression: (sin^4(x) - cos^4(x)) / (sin^2(x) - cos^2(x))

Simplify the trigonometric expression (sin^4(x) - cos^4(x)) / (sin^2(x) - cos^2(x)) using the difference of squares formula and the Pythagorean identity. This problem covers key concepts in trigonometry and algebra and is suitable for students in Grades 10-12.

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