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Let's break down and solve each problem based on the provided information:

### 1. Given: \( \tan A = \frac{\sqrt{7}}{21} \), \( 0^\circ < A < 90^\circ \)
**Find: \( \cos 2A \)**

We know the identity for \( \cos 2A \):
\[
\cos 2A = \frac{1 - \tan^2 A}{1 + \tan^2 A}
\]
Given \( \tan A = \frac{\sqrt{7}}{21} \), we can substitute this into the formula to find \( \cos 2A \).

### 2. Given: \( \tan A = -\frac{5}{15} = -\frac{1}{3} \), \( 90^\circ < A < 180^\circ \)
**Find: \( \cos 2A \)**

Since \( \tan A = -\frac{1}{3} \), we use the same formula as in the first problem:
\[
\cos 2A = \frac{1 - \tan^2 A}{1 + \tan^2 A}
\]
We will substitute \( \tan A = -\frac{1}{3} \) and simplify to find \( \cos 2A \).

### 3. Given: \( \sin A = -\frac{4}{9} \), \( 180^\circ < A < 270^\circ \)
**Find: \( \cos 2A \)**

We use the identity involving sine to find \( \cos 2A \):
\[
\cos 2A = 1 - 2 \sin^2 A
\]
Substitute \( \sin A = -\frac{4}{9} \) into this formula to find \( \cos 2A \).

### 4. Given: \( \cos A = \frac{\sqrt{65}}{65} \), \( 270^\circ < A < 360^\circ \)
**Find: \( \cos 2A \)**

We use the identity:
\[
\cos 2A = 2 \cos^2 A - 1
\]
Substitute \( \cos A = \frac{\sqrt{65}}{65} \) into the formula and simplify.

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Would you like further clarification or details on any of the formulas? Here are some related questions you might explore:

1. How do the double angle formulas for sine and cosine work?
2. What other identities are useful for solving problems with tangents?
3. How do we determine the sign of trigonometric functions in different quadrants?
4. How can we use the Pythagorean identity to find missing trigonometric values?
5. What are the common applications of these identities in real-world problems?

**Tip**: Always make sure to consider the angle's quadrant to assign the correct signs to the values of sine, cosine, and tangent.

1. Given: tan A = √7/21, 0° < A < 90°. Find: cos 2A
2. Given: tan A = -2/5, 90° < A < 180°. Find: cos 2A
3. Given: sin A = -4/9, 180° < A < 270°. Find: cos 2A
4. Given: cos A = √65/65, 270° < A < 360°. Find: cos 2A

Solve for cos 2A using given values for tan A, sin A, and cos A. Problems cover the double angle formula and determining trigonometric values based on angle quadrants. Suitable for Grades 11-12.

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