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

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

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

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

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

6. Given that \( \tan 4θ × \tan 6θ = 1 \) and \( 6θ \) is a positive acute angle, find the value of \( \tan 5θ \).

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

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

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

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

11. 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

12. 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) \(\cfrac{10}{13}\)

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

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

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

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

17. 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 \( \sec^2\theta = 1 + \tan^2\theta \).

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

19. 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) \(\cfrac{\sqrt5}{\sqrt2}\)

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

21. 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°

22. If \(a\) is a positive number and \[ a : \cfrac{27}{64} = \cfrac{3}{4} : a, \] then find the value of \(a\).

(a) \(\cfrac{81}{256}\) (b) 9 (c) \(\cfrac{9}{16}\) (d) \(\cfrac{16}{9}\)

23. If \(A + B = 90^\circ\) and \(\tan A = \frac{3}{4}\), then what is the value of \(\cot B\)?

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

24. If \(\tan^2 θ + \cot^2 θ = \frac{10}{3}\), then find the values of \(\tan θ + \cot θ\) and \(\tan θ - \cot θ\), and from there calculate the value of \(\tan θ\).

25. If \(\cot θ = 2\), then find the values of \(\tan θ\) and \(\sec θ\), and show that: \[1 + \tan^2θ = \sec^2θ\]

26. If \( \tan^2\theta + \cot^2\theta = \cfrac{10}{3} \), then find the values of \( \tan\theta + \cot\theta \) and \( \tan\theta - \cot\theta \). From there, determine the value of \( \tan\theta \).

27. If \( \tan\theta + \cot\theta = 2 \), then the value of \( \tan\theta - \cot\theta \) is —

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

28. If \( \tan\alpha + \cot\alpha = \sqrt{3} \), then what is the value of \( \tan^3\alpha + \cot^3\alpha \)?

29. If \( \cot\alpha = \tan(\beta + \gamma) \), then what is the value of \( \sin(\alpha + \beta + \gamma) \)?

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

30. If \( \tan θ + \cot θ = 2 \), then what is the value of \( \tan^7 θ + \cot^7 θ \)?