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To solve the problem:

**Problem Statement:**
Find the exact value of \( \sin \left( 2 \tan^{-1} \left( \frac{2}{5} \right) \right) \).

### Solution Steps:

1. **Let** \( \theta = \tan^{-1} \left( \frac{2}{5} \right) \). This means \( \tan \theta = \frac{2}{5} \).

2. **Using a Double-Angle Identity** for sine:
   \[
   \sin(2\theta) = 2 \sin \theta \cos \theta
   \]

3. **Find \( \sin \theta \) and \( \cos \theta \):**
   - Draw a right triangle where \( \tan \theta = \frac{2}{5} \), so the opposite side is 2, and the adjacent side is 5.
   - Use the Pythagorean theorem to find the hypotenuse \( h \):
     \[
     h = \sqrt{2^2 + 5^2} = \sqrt{4 + 25} = \sqrt{29}
     \]
   - Thus:
     \[
     \sin \theta = \frac{2}{\sqrt{29}} \quad \text{and} \quad \cos \theta = \frac{5}{\sqrt{29}}
     \]

4. **Apply the values in the double-angle formula**:
   \[
   \sin(2\theta) = 2 \cdot \frac{2}{\sqrt{29}} \cdot \frac{5}{\sqrt{29}} = 2 \cdot \frac{10}{29} = \frac{20}{29}
   \]

### Final Answer:
\[
\sin \left( 2 \tan^{-1} \left( \frac{2}{5} \right) \right) = \frac{20}{29}
\]

Would you like further explanation or have questions on any step?

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Here are five related questions to extend your understanding:

1. How would you use double-angle identities to find \( \cos(2\theta) \) if \( \tan \theta = \frac{2}{5} \)?
2. How can you derive the formula for the double-angle identity of sine?
3. What is the significance of the inverse tangent function in trigonometry?
4. How does the Pythagorean theorem help in solving trigonometric expressions involving right triangles?
5. What are other ways to express \( \sin(2\theta) \) using different trigonometric identities?

**Tip:** Double-angle identities are powerful tools in trigonometry and are especially useful in simplifying expressions involving angles and trigonometric functions.

Find the exact value of \( \sin \left( 2 \tan^{-1} \left( \frac{2}{5} \right) \right) \).

Learn how to find the exact value of \( \sin \left( 2 \tan^{-1} \left( \frac{2}{5} \right) \right) \) using double-angle identities and trigonometric functions. This solution includes steps for calculating \( \sin \theta \) and \( \cos \theta \) using a right triangle and the Pythagorean theorem. Suitable for high school students studying advanced trigonometry.

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