1. If \(\theta\) is a positive acute angle and \( \sin\theta = \cos(2\theta + 15^\circ) \), then what is the value of \(\theta\)?

(a) 30° (b) 25° (c) 60° (d) 90°

2. If \(\sin 3\theta = \frac{1}{2}\), then what is the value of \(\cos(80^\circ + \theta)\)?

3. If \(\sin (\theta + 30^\circ) = \cos 15^\circ\), then what is the value of \(\cos 2\theta\)?

4. If \( \tan^4\theta + \tan^2\theta = 1 \), then what is the value of \( \cos^4\theta + \cos^2\theta - 1 \)?

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

5. If \( \cos\theta - \sin\theta = \sqrt{2} \sin\theta \), then what is the value of \( \cos\theta + \sin\theta \)?

(a) \(2 cos\theta\) (b) \(\sqrt2 sin \theta\) (c) \(2 sin\theta\) (d) \(\sqrt2 cos \theta\)

6. 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) \(\cfrac{x^2-y^2}{x^2+y^2}\)

7. 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) \(-r^2\)

8. If \( \sin\theta \cos\theta = \frac{1}{2} \), then what is the value of \( (\sin\theta + \cos\theta)^2 \)?

(a) 1 (b) 3 (c) 2 (d) 4

9. If \(5\cos\theta + 12\sin\theta = 13\), then what is the value of \(\tan\theta\)?

(a) \(\cfrac{13}{15}\) (b) \(\cfrac{12}{5}\) (c) \(\cfrac{5}{13}\) (d) \(\cfrac{5}{12}\)

10. If \( \cos\theta + \sin\theta = \sqrt{2} \cos\theta \), then what is the value of \( \sin\theta - \cos\theta \)?

(a) \(\sqrt2 sin\theta\) (b) \(-\sqrt2 sin\theta\) (c) \(-\sqrt2 cos\theta\) (d) None of the above

11. If \( \sin\theta \cos\theta = \frac{1}{2} \), then what is the value of \( (\sin\theta + \cos\theta)^2 \)?

(a) 1 (b) 3 (c) 2 (d) 4

12. If \( a\cos\theta + b\sin\theta = c \), then what is the value of \( a\sin\theta - b\cos\theta \)?

(a) \(\pm\sqrt{a^2-b^2+c^2}\) (b) \(\pm\sqrt{a^2+b^2-c^2}\) (c) \(\pm\sqrt{a^2-b^2-c^2}\) (d) \(\pm\sqrt{b^2+c^2-a^2}\)

13. If \(\tan(\theta + 15^\circ) = \sqrt{3}\), then what is the value of \(\sin \theta\)?

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

15. If \(\frac{\sin \theta + \cos \theta}{\sin \theta - \cos \theta} = 5\), then what is the value of \(\tan \theta\)?

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

17. If \(\sin\theta \cos\theta = \frac{1}{2}\), then what is the value of \((\sin\theta - \cos\theta)^2\)?

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

18. If \(\tan(\theta + 15^\circ) = 1\), then what is the value of \(\cos 2\theta\)?

(a) \(\cfrac{1}{2}\) (b) \(\cfrac{1}{\sqrt{2}}\) (c) \(\cfrac{\sqrt{3}}{2}\) (d) \(1\)

19. If \(a \sin^2 \theta + b \cos^2 \theta = c\), then what is the value of \(\tan^2 \theta\)?

(a) \(\cfrac{b+c}{a+c}\) (b) \(\cfrac{b-c}{c-a}\) (c) \(\cfrac{a-b}{b-c}\) (d) \(\cfrac{c-a}{a-b}\)

20. If \(\cos\theta = \frac{4}{5}\), then what is the value of \(\csc\theta \div (1 + \cot\theta)\)?

(a) \(\cfrac{2}{4}\) (b) \(\cfrac{5}{7}\) (c) \(\cfrac{3}{5}\) (d) \(\cfrac{1}{3}\)

21. If \(\tan\theta = 3\), then what is the value of \(a\sin\theta + b\cos\theta\)?

(a) \(\sqrt{a^2+b^2}\) (b) \(\sqrt{a^2-b^2}\) (c) \(\sqrt{a^3+b^3}\) (d) \(\sqrt{a^3-b^3}\)

22. If \(\sin\theta + \cos\theta = \sqrt{2}\), then the value of \(\theta\) will be—

(a) \(\cfrac{\pi^c}{2}\) (b) \(\cfrac{\pi^c}{3}\) (c) \(\pi^c\) (d) \(\cfrac{\pi^c}{4}\)

23. If \(\cos\theta + \sec\theta = 2\), then what is the value of \(\cos^{11}\theta + \sec^{11}\theta\)?

(a) 0 (b) 1 (c) 2 (d) 22

24. If \(2\cos(3\theta) = 1\), then what is the value of \(\theta\)?

(a) \(10^\circ\) (b) \(15^\circ\) (c) \(20^\circ\) (d) \(30^\circ\)

25. If \( \tan\theta = \cfrac{4}{3} \), then what is the value of \( \sin\theta + \cos\theta \)?

26. If \( \sin\theta - \cos\theta = \frac{7}{13} \), then the value of \( \sin\theta + \cos\theta \) is —

(a) \(\cfrac{13}{17}\) (b) \(\cfrac{17}{13}\) (c) \(\cfrac{13}{7}\) (d) None of the above

27. If \( \cos^2\theta - \sin^2\theta = \frac{1}{2} \), then what is the value of \( \tan\theta \)?

(a) \(-\cfrac{1}{\sqrt3}\) (b) \(\cfrac{1}{3}\) (c) \(\cfrac{1}{\sqrt3}\) (d) \(-\cfrac{1}{3}\)

28. If \[ \frac{\sec \theta + \tan \theta}{\sec \theta - \tan \theta} = \frac{2 + \sqrt{3}}{2 - \sqrt{3}}, \] then what is the value of \( \theta \)?

(a) 60° (b) 30° (c) 45° (d) 90°

29. If \(\csc\theta + \cot\theta = 2 + \sqrt{3}\), then what is the value of \(\csc\theta - \cot\theta\)?

(a) \(\sqrt3+\sqrt2\) (b) \(\sqrt2-3\) (c) \(\sqrt3-2\) (d) \(2-\sqrt3\)

30. 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°