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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.

Learn how to simplify a trigonometric expression involving sine and cosine with detailed steps and identities. This problem covers key trigonometric concepts such as double-angle identities and their applications. Suitable for high school students.

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