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We are tasked with proving the following trigonometric identity:

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

### Step-by-Step Solution:

1. **Factor the left-hand side**:
   Notice that \(\sin(x)\) is common in both terms on the left-hand side, so we factor it out:
   \[
   \sin(x) \cos^2(x) - \sin(x) = \sin(x) \left( \cos^2(x) - 1 \right)
   \]

2. **Use a Pythagorean identity**:
   From the fundamental trigonometric identity:
   \[
   \sin^2(x) + \cos^2(x) = 1
   \]
   We can rearrange this as:
   \[
   \cos^2(x) - 1 = -\sin^2(x)
   \]

3. **Substitute**:
   Substituting \(\cos^2(x) - 1 = -\sin^2(x)\) into the factored expression:
   \[
   \sin(x) \left( \cos^2(x) - 1 \right) = \sin(x) \left( -\sin^2(x) \right)
   \]

4. **Simplify**:
   Now simplify the expression:
   \[
   \sin(x) \left( -\sin^2(x) \right) = -\sin^3(x)
   \]

Thus, we have shown that:
\[
\sin(x) \cos^2(x) - \sin(x) = -\sin^3(x)
\]

### Final Answer:
The identity is proven.

Do you have any questions or would you like further details?

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Here are 5 related questions to explore:

1. How do we derive the Pythagorean identity \(\sin^2(x) + \cos^2(x) = 1\)?
2. What are some other common trigonometric identities involving sine and cosine?
3. Can we prove this identity using a different method, such as substituting specific angle values?
4. How do trigonometric identities help in simplifying integrals and derivatives?
5. How does factoring expressions help in solving trigonometric equations?

**Tip:** Always look for common factors in algebraic or trigonometric expressions before simplifying. This often makes complex problems easier to handle!

Show that: sin(x)cos^2(x) - sin(x) = -sin^3(x)

Step-by-step solution to prove the trigonometric identity sin(x)cos^2(x) - sin(x) = -sin^3(x) using factoring and the Pythagorean identity. Suitable for students in Grades 10-12 studying trigonometry.

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