Great! Let's begin with 30 fundamental trigonometric manipulations. Here’s the first set of easier problems to help you build a strong foundation:

Problem Set 1: Basic Trigonometric Identities

1. Simplify: $\sin^2(x) + \cos^2(x)$
2. Simplify: $\frac{\sin(x)}{\cos(x)}$
3. Simplify: $1 + \tan^2(x)$
4. Express in terms of sine and cosine: $\cot(x)$
5. Verify: $\tan(2x) = \frac{2\tan(x)}{1 - \tan^2(x)}$
6. Simplify: $\sin(2x) \cdot \cos(2x)$
7. Express: $\sec(x)$ in terms of sine.
8. Find: $\cos(60^\circ)$
9. Find: $\sin(45^\circ)$
10. Simplify: $\frac{1 - \cos(2x)}{2}$

Problem Set 2: Expanding and Factoring

11. Expand: $(\sin(x) + \cos(x))^2$
12. Factor: $\sin^2(x) - \cos^2(x)$
13. Express: $\sin(2x)$ using double angle identities.
14. Verify: $\cos(2x) = 2\cos^2(x) - 1$
15. Simplify: $\sin(3x)$ in terms of $\sin(x)$

Problem Set 3: Combining Functions

16. Simplify: $\tan(x) \cdot \cot(x)$
17. Express: $1 - \sin^2(x)$ in terms of cosine.
18. Find: $\csc(x) \cdot \sin(x)$
19. Simplify: $\frac{\cos^2(x)}{\sec(x)}$
20. Simplify: $\tan(x + \frac{\pi}{4})$

Problem Set 4: Advanced Simplification and Verification

21. Verify: $\cos^2(x) - \sin^2(x) = \cos(2x)$
22. Simplify: $\frac{1}{\sin(x) \cdot \cot(x)}$
23. Express in simplest form: $\sec(x) \cdot \cos(x)$
24. Find: $\sin^2(x) \cdot \csc^2(x)$
25. Verify: $\frac{1}{1 + \tan^2(x)} = \cos^2(x)$

Problem Set 5: Involving Sum and Difference Identities

26. Simplify: $\sin(x + y) + \sin(x - y)$
27. Verify: $\cos(x + y) \cdot \cos(x - y) = \cos^2(x) - \sin^2(y)$
28. Express in terms of sine and cosine: $\tan(x + y)$
29. Simplify: $\cos(\pi - x)$
30. Simplify: $\sin(\frac{\pi}{2} - x)$

These cover a broad range of foundational trigonometric manipulations, from basic identities to sum and difference formulas. Try solving them step by step, and let me know if you'd like me to explain or verify your solutions.

Would you like me to guide you through any of these problems?

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5 Relative Questions

1. Why is $\sin^2(x) + \cos^2(x) = 1$ considered a fundamental identity?
2. How do the double angle formulas simplify trigonometric expressions?
3. What is the significance of verifying identities in trigonometry?
4. How can expressing everything in terms of sine and cosine simplify complex expressions?
5. Why is understanding sum and difference identities important in trigonometry?

Tip:

Always start by expressing everything in terms of sine and cosine when dealing with trigonometric identities. This makes it easier to manipulate and simplify expressions.