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

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

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

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

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

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

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

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

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

11. \(\theta\) 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°

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

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

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

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

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

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

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

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

20. If \(\sec \theta - \tan \theta = \frac{1}{2}\), then find the values of \(\sec \theta\) and \(\tan \theta\).

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 \( \tan 4θ \cdot \tan 6θ = 1 \), then determine the value of \( θ \) given that \( 0° < θ < 90° \).

(a) 5° (b) 4° (c) 9° (d) 3°

23. If \(\sec θ - \tan θ = \frac{1}{\sqrt{3}}\), then find the values of both \(\sec θ\) and \(\tan θ\).

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 \( 2x = \sec A \) and \( \cfrac{2}{x} = \tan A \), then find the value of \( 2\left(x^2 - \cfrac{1}{x^2}\right) \).

(a) \(\cfrac{1}{2}\) (b) \(\cfrac{1}{4}\) (c) \(\cfrac{1}{8}\) (d) \(\cfrac{1}{16}\)

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

27. 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 \( \tan\theta \).

28. 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

29. In a circle with center \(O\), \(\bar{AB}\) is a diameter. On the opposite side of the circumference from the diameter \(\bar{AB}\), there are two points \(C\) and \(D\) such that \(\angle AOC = 130°\) and \(\angle BDC = x°\). Find the value of \(x\).

(a) 25° (b) 50° (c) 60° (d) 65°

30. 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