The equation \(\sin(4x) = 2\sin(2x)\cos(2x)\) can be derived using a trigonometric identity. Specifically, this uses the **double-angle identity for sine**, which states:

\[
\sin(2\theta) = 2\sin(\theta)\cos(\theta)
\]

### Explanation:

1. **Start with the left-hand side: \(\sin(4x)\)**:
   
   We can express \(4x\) as \(2 \cdot 2x\). Thus, we rewrite \(\sin(4x)\) as \(\sin(2(2x))\).

2. **Apply the double-angle identity:**
   
   According to the double-angle identity:
   \[
   \sin(2A) = 2\sin(A)\cos(A)
   \]
   If we set \(A = 2x\), we get:
   \[
   \sin(4x) = \sin(2(2x)) = 2\sin(2x)\cos(2x)
   \]

Thus, we have shown that:
\[
\sin(4x) = 2\sin(2x)\cos(2x)
\]

### Conclusion:
The expression \(\sin(4x)\) is indeed equal to \(2\sin(2x)\cos(2x)\) by applying the double-angle formula.

Would you like a more detailed breakdown or have any other questions?

Here are 5 related questions that might interest you:
1. How can you derive other trigonometric double-angle identities?
2. What is the double-angle identity for \(\cos(2x)\)?
3. Can we express \(\cos(4x)\) in terms of \(\cos(2x)\) and \(\sin(2x)\)?
4. How do trigonometric identities help simplify integrals?
5. What is the relationship between double-angle and half-angle identities?

**Tip:** Remember that mastering fundamental trigonometric identities can make solving more complex problems much easier.

Discover how to prove the trigonometric identity sin(4x) = 2sin(2x)cos(2x) using the double-angle formula. This problem is ideal for high school students learning advanced trigonometry concepts. Step-by-step explanation provided.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation