Given: tan A = √7/21,     0° < A < 90°
Find: cos 2A

We are given \( \tan A = \frac{\sqrt{7}}{21} \) and \( 0^\circ < A < 90^\circ \). We need to find \( \cos 2A \).

### Steps:

1. **Use the identity for \( \cos 2A \):**
   \[
   \cos 2A = \frac{1 - \tan^2 A}{1 + \tan^2 A}
   \]
   First, calculate \( \tan^2 A \):
   \[
   \tan^2 A = \left( \frac{\sqrt{7}}{21} \right)^2 = \frac{7}{441} = \frac{1}{63}
   \]

2. **Substitute into the identity for \( \cos 2A \):**
   \[
   \cos 2A = \frac{1 - \frac{1}{63}}{1 + \frac{1}{63}} = \frac{\frac{63}{63} - \frac{1}{63}}{\frac{63}{63} + \frac{1}{63}} = \frac{\frac{62}{63}}{\frac{64}{63}} = \frac{62}{64} = \frac{31}{32}
   \]

Thus, \( \cos 2A = \frac{31}{32} \).

Would you like further details on any of these steps or clarifications?

Here are some related questions to explore:
1. How do you derive the identity for \( \cos 2A \) using trigonometric identities?
2. What is the general relation between tangent and cosine functions?
3. Can we find \( \sin 2A \) using a similar approach?
4. How does \( \tan A \) relate to the sides of a right triangle?
5. What are the possible applications of double angle identities in real-world problems?

**Tip:** Always remember that the identity \( \cos 2A \) has multiple forms, like \( \cos^2 A - \sin^2 A \), which may be useful depending on the given data.

Given: tan A = √7/21, 0° < A < 90° Find: cos 2A

Learn how to find cos 2A when tan A = √7/21 for 0° < A < 90° using double angle identities in trigonometry. This problem covers key concepts like double angle formulas and the relation between tangent and cosine functions. Suitable for high school students in Grades 10-12.

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