1. If \(x = a \cos\theta + b \sin\theta\) and \(y = a \sin\theta - b \cos\theta\), then prove that \(x^2 + y^2 = a^2 + b^2\).

2. If \( \cosθ = \frac{x}{\sqrt{x^2 + y^2}} \), then prove that \( x\sinθ = y\cosθ \).

3. If \( \cot\theta = \frac{x}{y} \), then prove that \[ \frac{x\cos\theta - y\sin\theta}{x\cos\theta + y\sin\theta} = \frac{x^2 - y^2}{x^2 + y^2} \]

4. If \(\cfrac{\sin \theta}{x} = \cfrac{\cos \theta}{y}\), then prove that \( \sin \theta - \cos \theta = \cfrac{x - y}{\sqrt{x^2 + y^2}} \).

5. If \(\cfrac{\sin\theta}{x} = \cfrac{\cos\theta}{y}\), then prove that \(\sin\theta - \cos\theta = \cfrac{x - y}{\sqrt{x^2 + y^2}}\).

6. If \(x = r\cos\theta\cos\phi\), \(y = r\cos\theta\sin\phi\), and \(z = r\sin\theta\), then what is the value of \(x^2 + y^2 + z^2\)?

(a) \(r\) (b) \(1\) (c) \(r^2\) (d) \(-r^2\)

7. If \(2y \cos \theta = x \sin \theta\) and \(2x \sec \theta - y \cosec \theta = 3\), then \(x^2 + 4y^2 = ?\)

(a) 2 (b) 1 (c) 0 (d) None of the above

8. If \[ \frac{\sinθ}{x} = \frac{\cosθ}{y} \] then prove that \[ \sinθ - \cosθ = \frac{x - y}{\sqrt{x^2 + y^2}} \]

9. If \(x = 3 \cos \theta\) and \(y = 3 \sin \theta\), then what is the value of \(x^2 + y^2\)?

10. If \(a(\tan\theta + \cot\theta) = 1\) and \(\sin\theta + \cos\theta = b\), then prove that \(2a = b^2 - 1\), where \(0^\circ < \theta < 90^\circ\).

11. If \(x = a\cos(90^\circ - \theta)\), \(y = b\cot(90^\circ - \theta)\), then prove that \(\cfrac{a^2}{x^2} - \cfrac{b^2}{y^2} = 1\).

12. If \(x = \frac{1 + \sin\theta}{\cos\theta}\), then show that \(x^2 - 2x\tan\theta - 1 = 0\).

13. If \( \alpha \) and \( \beta \) are complementary angles, then prove that \[ \cot \beta + \cos \beta = \frac{\cos \beta}{\cos \alpha (1 + \sin \beta)} \]

14. If \( \cos\theta + \sqrt{3} \sin\theta = 2 \sin\theta \), then prove that \( \sin\theta - \sqrt{3} \cos\theta = 2 \cos\theta \).

15. Prove that \[ \cfrac{\tan\theta + \sec\theta - 1}{\tan\theta - \sec\theta + 1} = \cfrac{1 + \sin\theta}{\cos\theta} \] If you'd like, I can help walk through the proof step-by-step as well. Ready when you are!

16. If \( \tan \theta = \frac{x}{y} \), then what is the value of \[ \frac{x\sin\theta - y\cos\theta}{x\sin\theta + y\cos\theta}? \]

(a) \(\cfrac{x^2+y^2}{x^2-y^2}\) (b) \(\cfrac{x-y}{x+y}\) (c) \(\cfrac{x+y}{x-y}\) (d) \(\cfrac{x^2-y^2}{x^2+y^2}\)

17. If \(\sin(3x - 20^\circ) = \cos(3y + 20^\circ)\), then what is the value of \(x + y\)?

(a) 60° (b) 30° (c) 45° (d) 90°

18. If \(∠P + ∠Q = 90^\circ\), then prove that \[ \sqrt{\frac{\sin P}{\cos Q}} - \sin P \cos Q = \cos^2 P \]

19. If \(\cos 52^\circ = \frac{x}{\sqrt{x^2 + y^2}}\), then what is the value of \(\tan 38^\circ\)?

20. If \(x = cy + bz\), \(y = az + cx\), and \(z = bx + ay\), then prove that \[ \frac{x^2}{1 - a^2} = \frac{y^2}{1 - b^2} \]

21. If \(\cos 43^\circ = \frac{x}{\sqrt{x^2 + y^2}}\), then what is the value of \(\tan 47^\circ\)?

22. If \(\sin^4 \theta + \sin^2 \theta = 1\), then prove that \(\tan^4 \theta - \tan^2 \theta = 1\).

23. If \(x = cy + bz\), \(y = az + cx\), and \(z = bx + ay\), then prove that \[ \frac{x^2}{1 - a^2} = \frac{y^2}{1 - b^2} = \frac{z^2}{1 - c^2} \]

24. If \(\tan\theta = \frac{x}{y}\), then what is the value of \[ \frac{x\sin\theta - y\cos\theta}{x\sin\theta + y\cos\theta}? \]

(a) \(\cfrac{x^2-y^2}{x^2+y^2}\) (b) \(\cfrac{y^2-x^2}{x^2+y^2}\) (c) \(\cfrac{x^2+y^2}{y^2-x^2}\) (d) None of the above

25. If \(\cot \theta = \cfrac{x}{y}\), then the value of \(\cfrac{x\cos\theta - y\sin\theta}{x\cos\theta + y\sin\theta}\) is —

(a) \(\cfrac{x^2+y^2}{x^2-y^2}\) (b) \(\cfrac{x^2}{x^2-y^2}\) (c) \(\cfrac{x^2}{x^2-y^2}\) (d) \(\cfrac{x^2-y^2}{x^2+y^2}\)

26. If \( \tan\theta = \frac{x}{y} \), then find the value of \[ \frac{x \sin\theta - y \cos\theta}{x \sin\theta + y \cos\theta} \]

27. If \(x = a\sinθ\) and \(y = b\tanθ\), then prove that \(\cfrac{a^2}{x^2} - \cfrac{b^2}{y^2} = 1\)

28. If \(x = r\sinθ \cosϕ\), \(y = r\sinθ \sinϕ\), and \(z = r\cosθ\), then find the value of \(x^2 + y^2 + z^2\).

(a) \(r\) (b) \(5r\) (c) \(\sqrt{r}\) (d) \(r^2\)

29. If \(\sin^2\theta + \sin^4\theta = 1\), then prove that \(\tan^4\theta - \tan^2\theta = 1\).

30. If \(\sin\theta+\sin^2\theta=1\), prove that \(\cos^2\theta+\cos^4\theta=1\).