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The expression given is:

\[
\sin^3(x) - \cos^3(x)
\]

This is a difference of cubes, which can be factored using the identity:

\[
a^3 - b^3 = (a - b)(a^2 + ab + b^2)
\]

Here, \(a = \sin(x)\) and \(b = \cos(x)\), so applying the identity, we get:

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

We know that \(\sin^2(x) + \cos^2(x) = 1\), so the expression simplifies further:

\[
\sin^3(x) - \cos^3(x) = (\sin(x) - \cos(x))(1 + \sin(x)\cos(x))
\]

Thus, the factored and simplified form of the expression is:

\[
(\sin(x) - \cos(x))(1 + \sin(x)\cos(x))
\]

Would you like more details or have any further questions?

Here are 5 related questions to explore:
1. How do you factor the sum of cubes, \(a^3 + b^3\)?
2. What is the geometric interpretation of the sine and cosine functions?
3. How can you use trigonometric identities to simplify other expressions?
4. What are some real-world applications of the sine and cosine functions?
5. How does the Pythagorean identity \(\sin^2(x) + \cos^2(x) = 1\) apply in other trigonometric problems?

**Tip:** Always look for identities, such as the Pythagorean identity, when simplifying trigonometric expressions!

Factor the trigonometric expression sin^3(x) - cos^3(x) and simplify.

Learn how to factor and simplify the trigonometric expression sin^3(x) - cos^3(x) using the difference of cubes identity and the Pythagorean identity. This problem is suitable for students in Grades 10-12.

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