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To solve the given problem:

**Problem:**  
Given: \(\tan A = \frac{\sqrt{7}}{2}\), \(0^\circ < A < 90^\circ\)  
Find: \(\cos 2A\)

**Solution:**

1. **Find \(\sin A\) and \(\cos A\) using \(\tan A\):**  
   Since \(\tan A = \frac{\sin A}{\cos A} = \frac{\sqrt{7}}{2}\), we can let \(\sin A = \sqrt{7}k\) and \(\cos A = 2k\) for some constant \(k\).

2. **Using the Pythagorean identity:**  
   \[
   \sin^2 A + \cos^2 A = 1
   \]  
   Substitute \(\sin A\) and \(\cos A\):  
   \[
   (\sqrt{7}k)^2 + (2k)^2 = 1  
   \]  
   \[
   7k^2 + 4k^2 = 1  
   \]  
   \[
   11k^2 = 1  
   \]  
   \[
   k^2 = \frac{1}{11}  
   \]  
   \[
   k = \frac{1}{\sqrt{11}}  
   \]

3. **Find \(\sin A\) and \(\cos A\):**  
   \[
   \sin A = \sqrt{7} \cdot \frac{1}{\sqrt{11}} = \frac{\sqrt{7}}{\sqrt{11}} = \frac{\sqrt{77}}{11}  
   \]  
   \[
   \cos A = 2 \cdot \frac{1}{\sqrt{11}} = \frac{2}{\sqrt{11}} = \frac{2\sqrt{11}}{11}  
   \]

4. **Calculate \(\cos 2A\) using the double angle formula:**  
   \[
   \cos 2A = \cos^2 A - \sin^2 A  
   \]  
   Substitute the values of \(\sin A\) and \(\cos A\):  
   \[
   \cos 2A = \left(\frac{2\sqrt{11}}{11}\right)^2 - \left(\frac{\sqrt{77}}{11}\right)^2  
   \]  
   \[
   \cos 2A = \frac{4 \cdot 11}{121} - \frac{77}{121}  
   \]  
   \[
   \cos 2A = \frac{44}{121} - \frac{77}{121}  
   \]  
   \[
   \cos 2A = \frac{44 - 77}{121}  
   \]  
   \[
   \cos 2A = \frac{-33}{121} = -\frac{33}{121}  
   \]

Therefore, \(\cos 2A = -\frac{33}{121}\).

Would you like more details or have any other questions?

Here are some related questions for further exploration:
1. How can you find \(\sin 2A\) using \(\sin A\) and \(\cos A\)?
2. What is the general formula for \(\tan 2A\) in terms of \(\tan A\)?
3. How can \(\cos 2A\) be expressed using only \(\cos A\)?
4. Can you derive \(\cos 2A\) using the half-angle identity?
5. What happens to \(\cos 2A\) when \(\tan A\) is negative?

**Tip:** Remember, the double angle identities can simplify many trigonometric expressions, especially when solving for angles or finding specific values.

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

Solve the trigonometric problem where tan A = √7/2 and 0° < A < 90°. Learn how to find cos 2A using trigonometric identities like the Pythagorean identity and the double angle formula. Suitable for Grades 10-12.

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