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?

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!