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To solve for the exact value of \( \cos \left( 2 \sin^{-1} \frac{5}{8} \right) \), we can proceed as follows:

1. **Let \( \theta = \sin^{-1} \frac{5}{8} \):**
   This implies that \( \sin \theta = \frac{5}{8} \).

2. **Find \( \cos \theta \):**
   Since \( \sin^2 \theta + \cos^2 \theta = 1 \), we have:
   \[
   \cos^2 \theta = 1 - \sin^2 \theta = 1 - \left(\frac{5}{8}\right)^2 = 1 - \frac{25}{64} = \frac{39}{64}
   \]
   Therefore, \( \cos \theta = \sqrt{\frac{39}{64}} = \frac{\sqrt{39}}{8} \).

3. **Use the Double Angle Formula for Cosine:**
   We need to find \( \cos(2\theta) \). Using the double angle identity:
   \[
   \cos(2\theta) = 2\cos^2 \theta - 1
   \]
   Substitute \( \cos \theta = \frac{\sqrt{39}}{8} \):
   \[
   \cos(2\theta) = 2 \left(\frac{\sqrt{39}}{8}\right)^2 - 1 = 2 \cdot \frac{39}{64} - 1 = \frac{78}{64} - 1 = \frac{78 - 64}{64} = \frac{14}{64} = \frac{7}{32}
   \]

4. **Final Answer:**
   \[
   \cos \left( 2 \sin^{-1} \frac{5}{8} \right) = \frac{7}{32}
   \]

Would you like more details or have any questions on these steps?

### Related Questions

1. How would you find \( \sin(2\theta) \) instead of \( \cos(2\theta) \) given \( \sin \theta = \frac{5}{8} \)?
2. How do inverse trigonometric functions relate to triangle-based identities?
3. What are some other ways to verify trigonometric identities like the double angle identity?
4. How would the problem change if we were asked to find \( \cos \left( \sin^{-1} \frac{5}{8} \right) \) instead of \( 2 \sin^{-1} \frac{5}{8} \)?
5. What are other methods to simplify expressions involving inverse trigonometric functions?

**Tip:** Remember that using trigonometric identities can simplify complex expressions involving angles and inverse trigonometric functions.

Find the exact value of \( \cos \left( 2 \sin^{-1} \frac{5}{8} \right) \).

Learn how to find the exact value of \( \cos \left( 2 \sin^{-1} \frac{5}{8} \right) \) using inverse trigonometric functions and trigonometric identities. This solution covers key concepts such as the double angle formula for cosine and Pythagorean identity, ideal for students in Grades 10-12.

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