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

2. 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

3. What is the value of \(\cfrac{9}{\csc^2\theta} + 4\cos^2\theta + \cfrac{5}{1 + \tan^2\theta}\)?

(a) 1 (b) 0 (c) 3 (d) 9

4. If \( \tan\left(\cfrac{\pi}{2} - \cfrac{\alpha}{2}\right) = \sqrt{3} \), then what is the value of \( \cos\alpha \)?

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

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

6. If \( \sin^2A + \sin^4A = 1 \), then what is the value of \( \tan^2A - \tan^4A \)?

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

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

(a) \(-\cfrac{1}{\sqrt3}\) (b) \(\cfrac{1}{3}\) (c) \(\cfrac{1}{\sqrt3}\) (d) \(-\cfrac{1}{3}\)

8. If \[ \frac{\sec \theta + \tan \theta}{\sec \theta - \tan \theta} = \frac{2 + \sqrt{3}}{2 - \sqrt{3}}, \] then what is the value of \( \theta \)?

(a) 60° (b) 30° (c) 45° (d) 90°

9. If \(x\sin45^\circ \cos45^\circ \tan60^\circ = \tan^2 45^\circ - \cos^2 60^\circ\), then what is the value of \(x\)?

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

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. If \( \csc\theta \tan\theta = 2 \), then what is the value of \( \theta \)?

(a) 60° (b) 45° (c) 30° (d) 90°

12. What is the value of \( \sec^2 12^\circ - \frac{1}{\tan^2 78^\circ} \)?

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

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

14. If \( \cos^2\theta – \sin^2\theta = \frac{1}{2} \), then what is the value of \( \tan\theta \)?

(a) \(\frac{1}{\sqrt3}\) (b) \(\sqrt3\) (c) 1 (d) None of the above

15. \(\theta\) 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°

16. What is the value of \( \tan 35^\circ \cdot \tan 40^\circ \cdot \tan 45^\circ \cdot \tan 50^\circ \cdot \tan 55^\circ \)?

(a) \(\cfrac{1}{\sqrt3}\) (b) \(\sqrt3\) (c) 1 (d) 0

17. If \( \sin 51^\circ = \cfrac{a}{\sqrt{a^2 + b^2}} \), then what is the value of \( \tan 51^\circ + \tan 39^\circ \)?

(a) \(\cfrac{a^2+b^2}{ab}\) (b) \(\cfrac{a^2-b^2}{ab}\) (c) \(\cfrac{a-b}{ab}\) (d) \(\cfrac{a+b}{ab}\)

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

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

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

20. What is the value of \( \tan 1^\circ \times \tan 2^\circ \times \tan 3^\circ \times \ldots \times \tan 89^\circ \)?

21. What is the value of \(\cfrac{1}{\tan\theta+\cfrac{1}{\tan\theta}}\)?

(a) \(tan\theta\) (b) \(sin 2\theta\) (c) \(cos \theta\) (d) \(sin\theta cos\theta\)

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

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

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

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

26. If \(\sec^2 \theta + \tan^2 \theta = \frac{13}{12}\), then what is the value of \(\sec^4 \theta - \tan^4 \theta\)?

27. If \(\sin 3x = 1\), what is the value of \(\tan 2x\)?

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

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

30. 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) \(1\)