1. 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) \(1\)

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

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

3. If the roots of the equation \(ax^2 + b + c = 0\) are \(\sin α\) and \(\cos α\), then what is the value of \(b^2\)?

(a) \(a^2-2ac\) (b) \(a^2+2ac\) (c) \(a^2-ac\) (d) \(a^2+ac\)

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

5. 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°

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

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

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

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

10. If \(\cos^4\theta - \sin^4\theta = \frac{2}{3}\), then what is the value of \(1 - 2\sin^2\theta\)?

11. 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°

12. 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) \(\cfrac{1}{3}\)

13. 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

14. 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

15. 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) \(\sqrt2 cos \theta\)

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 \(x = r\cos\theta\cos\phi\), \(y = r\cos\theta\sin\phi\), 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) \(-r^2\)

18. 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

19. 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) \(\cfrac{5}{12}\)

20. 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

21. 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

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

23. If \(\frac{\sinθ + \cosθ}{\sinθ - \cosθ} = 7\), then what is the value of \(\tanθ\)?

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

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

26. If \(\theta\) is a positive acute angle and \(\sin \theta = \frac{\sqrt{3}}{2}\), then what is the value of \(\tan(\theta - 15^\circ)\)?

27. 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

28. 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

29. If \(\sin A + \sin B = 2\), where \(0^\circ \leq A \leq 90^\circ\) and \(0^\circ \leq B \leq 90^\circ\), then find the value of \(\cos A + \cos B\).

30. If \(3\sinθ + 4\cosθ = 5\), then what is the value of \(4\sinθ - 3\cosθ\)?