The image shows a trigonometry problem where you're asked to simplify an expression involving trigonometric identities. The problem is:

Given the following identities: $\sin(2x) = 2\sin(x)\cos(x) \quad \text{and} \quad \tan(x) = \frac{\sin(x)}{\cos(x)},$ simplify the expression involving $a$, $b$, and $c$: $a\sin(2x) - \cos(2x)\sin^2(x).$

You're then given the expression: $a\sin(2x) - \cos(2x)\sin^2(x) = \underline{\hspace{3cm}} \quad \text{(simplify your answer)}$

Simplifying the Expression:

1. Substitute the identities into the expression: $a \sin(2x) = a \cdot 2 \sin(x) \cos(x),$ so the expression becomes: $2a \sin(x) \cos(x) - \cos(2x) \sin^2(x).$

2. Use the identity for $\cos(2x)$. There are several forms of this identity, but one useful form is: $\cos(2x) = \cos^2(x) - \sin^2(x).$ Substituting this in: $2a \sin(x) \cos(x) - (\cos^2(x) - \sin^2(x)) \sin^2(x).$

3. Distribute $\sin^2(x)$ through the parentheses: $2a \sin(x) \cos(x) - \cos^2(x)\sin^2(x) + \sin^4(x).$

This expression can be simplified further, depending on the given values of $a$, $b$, and $c$, or possibly by recognizing patterns that could lead to a more compact form.

Would you like further details on simplifying this expression, or have any questions about the process?

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Here are some related questions to explore:

1. What are other forms of the double-angle identity for $\cos(2x)$?
2. How would the expression change if $a = b = c = 1$?
3. Can you prove the identity $\cos(2x) = 2\cos^2(x) - 1$?
4. How does the identity for $\tan(x)$ relate to $\sin(x)$ and $\cos(x)$?
5. What are the possible maximum and minimum values for $\sin(2x)$ and $\cos(2x)$?

Tip: When simplifying trigonometric expressions, always consider if an identity could simplify part of the expression first before multiplying out or expanding terms.