https://r2.erweima.ai/mathbot/image/1726189607394p8jeweon.pdf

The document you uploaded includes several trigonometric exercises and identities. Here’s a breakdown of the first set of exercises for clarity:

### Exercises:

1. **Calculate the trigonometric ratios** for the given angles based on the standard angles (30°, 45°, 60°):
   - \( \tan(16^\circ) \)
   - \( \tan(750^\circ) \)
   - \( \tan(225^\circ) \)
   - \( \tan(-105^\circ) \)
   - \( \tan(285^\circ) \)
   - \( \tan(-195^\circ) \)

2. **Find** \( \tan(a + \beta) \) and \( \tan(a - \beta) \) given that \( \tan(a) = \) and \( \tan(\beta) = \).

3. **Calculate**:
   - \( \tan(u + u) \)
   - \( \tan(u - u) \)
   - \( \cot(u + u) \)
   - \( \cot(u - u) \)
   - \( \sec(u + u) \)
   - \( \sec(u - u) \)

4. If \( \sin(a) = \tan(a) \), **find**:
   - \( \sin(a + h) \)
   - \( \sin(a + h) - \sin(a) \)
   - \( 40 \) (Not clear, might need more context.)

Let me know if you'd like detailed solutions to any of these exercises or if you'd like to explore specific problems.

### Expanded Questions:
1. What are the steps to compute the exact value of \( \tan(225^\circ) \)?
2. How do you simplify trigonometric expressions involving double-angle identities?
3. What is the method to verify trigonometric identities using a calculator?
4. How can you derive the sum and difference formulas for tangent?
5. What is the exact value of \( \sec(22.5^\circ) \)?

### Tip:
Remember, for angles greater than 360° or negative angles, reduce them to a reference angle within 0° to 360° using modulo 360.

Do you want more details or have any questions on this?

Solve various trigonometric exercises involving angles such as 16°, 225°, and 750°, and explore advanced topics like double angle identities and sum and difference formulas for tangent. This guide is designed for Grades 10-12 students.

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