1. If \( \tan \theta \cos 60° = \cfrac{√3}{2} \), find the value of \(\sin(\theta - 15°)\)

(a) \(\cfrac{1}{√2}\) (b) 1 (c) √2 (d) 0

2. If \(\sin(2x + y) = \cos(4x - y)\), find the value of \(\tan 3x\).

3. Given that \(\cos\alpha = \sin\beta\) and both \(\alpha\) and \(\beta\) are acute angles, find the value of \(\sin(\alpha + \beta)\).

4. If \(2 \cos^2\theta + 3 \sin \theta = 3\) and \(0^\circ < \theta < 90^\circ\), find the value of \(\theta\).

5. 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 \(\cos A + \cos B\).

6. If \(2\cos3\theta=1\), find the value of \(\theta\).

(a) 10° (b) 15° (c) 20° (d) 30°

7. Given that \(\sin 10\theta = \cos 8\theta\) and \(10\theta\) is a positive acute angle, find the value of \(\tan 9\theta\).

8. If \(\tan 4\theta \times \tan 6\theta = 1\) and \(6\theta\) is a positive acute angle, find the value of \(\theta\).

9. If \(\sin θ + \cos θ = 1\), then find the value of \(\sin θ × \cos θ\).

10. If \(\sin θ - \cos θ = \frac{7}{13}\), then find the value of \(\sin θ + \cos θ\).

11. If \(\sin(A + B) = 1\) and \(\cos(A - B) = 1\), then find the value of \(\cot 2A\), given that \(0^\circ \leq (A + B) \leq 90^\circ\) and \(A \geq B\).

12. If \(0^\circ \le \theta \le 90^\circ\) and \(3 - 3\sin\theta - \cos^2\theta = 0\), then find the value of \(\theta\).

(a) \(30^o\) (b) \(60^o\) (c) \(90^o\) (d) \(45^o\)

13. If \(\sin \theta + \sin^2 \theta = 1\), then find the value of \(\cos^2 \theta + \cos^4 \theta =\) _____

14. If \( \cos^4\theta - \sin^4\theta = \frac{2}{3} \), find the value of \( 1 - 2\sin^2\theta \).

15. Given: \[ r\cos\theta = 2\sqrt{3}, \quad r\sin\theta = 2 \] and \(\theta\) is an acute angle. Find the values of \(r\) and \(\theta\).

16. If \(2\cos(3\theta) = 1\), find the value of \(\theta\).

(a) 10° (b) 15° (c) 20° (d) 30°

17. If \(\cos^4\theta - \sin^4\theta = \cfrac{2}{3}\), then find the value of \(1 - 2\sin^2\theta\).

18. If \(\cfrac{\sinθ + \cosθ}{\sinθ - \cosθ} = \cfrac{3}{2}\), find the value of \(\cosθ\)

(a) \(\cfrac{1}{5}\) (b) \(\cfrac{3}{2}\) (c) \(\cfrac{1}{\sqrt{26}}\) (d) None of these

19. 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

20. 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°

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

22. If \(r\cosθ = 2\sqrt{3}\), \(r\sinθ = 2\), and \(0° < θ < 90°\), then find the values of \(r\) and \(θ\).

23. Given: \(\sin 5A = \csc (A + 36^\circ)\) and \(5A\) is a positive acute angle. Find the value of \(A\).

24. If \(r\cosθ = 1\) and \(r\sinθ = \sqrt{3}\), then find the values of \(r\) and \(θ\).

25. 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) \(\cfrac{\pi}{4}\)

26. If \( \cos^2 θ - \sin^2 θ = \frac{1}{2} \), then find the value of \( \cos^4 θ - \sin^4 θ \).

27. If \( \tan(θ + 15^\circ) = \sqrt{3} \), then find the value of \( \sinθ + \cosθ \).

28. If \( \cos^2 θ - \sin^2 θ = \cfrac{1}{2} \), then find the value of \( \tan^2 θ \).

29. If \( \sin10θ = \cos8θ \) and \(10θ\) is a positive acute angle, find the value of \( \tan9θ \).

30. From the equation \(5 \sin^2 \theta + 4 \cos^2 \theta = \frac{9}{2}\), find the value of \(\tan \theta\), where \(0^\circ < \theta < 90^\circ\).