1. \(\theta\) is a positive acute angle, and if \( \tan\theta = \cot\theta \), then what is the value of \(\theta\)?

(a) 40° (b) 45° (c) 60° (d) 20°

2. 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°

3. If \(\theta\) is a positive acute angle and \( \sin \theta - \cos \theta = 0 \), then the value of \(\cot 2\theta\) is –

(a) \(\cfrac{1}{√3}\) (b) 1 (c) √3 (d) 0

4. In a right-angled triangle, the two acute angles are \(\theta\) and \(\phi\). If \( \tan\theta = \cfrac{5}{12} \), then what is the value of \( \sin\phi \)?

(a) \(\cfrac{12}{13}\) (b) \(\cfrac{5}{13}\) (c) \(\cfrac{1}{4}\) (d) \(\cfrac{10}{13}\)

5. If \(tan4\theta \cdot tan6\theta = 1\) and \(6\theta\) is a positive acute angle, then determine the value of \(\theta\).

6. If \(sec 3\theta = cosec 2\theta\) and \(3\theta\) is a positive acute angle, find the value of \(\theta\).

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 \( \sec 5\theta = \csc(\theta + 36^\circ) \) and \(5\theta\) is a positive acute angle, then find the value of \( \theta \)."**

10. In a right-angled triangle, if the ratio of the perpendicular (opposite side) to the hypotenuse with respect to a positive acute angle \(\theta\) is \(12 : 13\), then determine the ratio of the perpendicular to the base and the ratio of the hypotenuse to the base, and verify that \( \sec^2\theta = 1 + \tan^2\theta \).

11. If \( \tan 4\theta \cdot \tan 6\theta = 1 \) and \(6\theta\) is a positive acute angle, then find the value of \( \tan 5\theta \).

12. Given \( \sin5\theta = \cos4\theta \) and \( 5\theta \) is a positive acute angle, what is the value of \( \tan3\theta \)?

13. If \(x\) is a real positive number and \(\sin x = \frac{2}{3}\), then what is the value of \(\tan x\)?

(a) \(\cfrac{2}{\sqrt5}\) (b) \(\cfrac{\sqrt5}{2}\) (c) \(\sqrt{\cfrac{5}{3}}\) (d) \(\cfrac{\sqrt5}{\sqrt2}\)

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

15. If \( \tan 2A = \cot (A - 18^\circ) \) and \(2A\) is a positive acute angle, then find the value of \(A\).

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

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

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

(a) \(5°\) (b) \(10°\) (c) \(9°\) (d) \(4°\)

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

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

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

22. 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) \(-r^2\)

23. In triangle ABC, \(\angle C = 90^\circ\) and AC : BC = 3 : 4, then what is the value of cosec A?

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

24. In triangle \( \triangle ABC \), if \( \angle B \) is a right angle and \( BC = \sqrt{3} \times AB \), then what is the value of \( \sin C \)?

(a) \(\frac{1}{2}\) (b) \(\frac{1}{\sqrt2}\) (c) \(\frac{\sqrt3}{2}\) (d) 1

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

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

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

28. In triangle ABC, let X and Y be the midpoints of sides AB and AC respectively. If \(BC + XY = 12\) units, then what is the value of \(BC - XY\)?

29. 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

30. In triangle ABC, ∠B = 90°, and BC = \(\sqrt{3}\) × AB. What is the value of \(\sin C\)?

(a) \(\cfrac{1}{2}\) (b) 1 (c) \(\cfrac{1}{\sqrt3}\) (d) \(\sqrt3\)