1. 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

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

3. The value of \( \tan1^\circ \cdot \tan2^\circ \cdot \tan3^\circ \cdots \tan88^\circ \cdot \tan89^\circ \) is

(a) 4 (b) 3 (c) 2 (d) 1

4. The value of \[ \tan 1^\circ \cdot \tan 2^\circ \cdot \tan 3^\circ \cdot \ldots \cdot \tan 89^\circ \] is \(1\).

5. Evaluate the value of: \[ \frac{\sin 43^\circ \cdot \sin 65^\circ \cdot \tan 25^\circ + \sin 65^\circ \cdot \cot 65^\circ \cdot \cos 47^\circ}{\cos 65^\circ \cdot \sqrt{1 - \cos^2 43^\circ}} \]

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

7. Find the value of \(x\) from the equation: \(x \sec^2 45^\circ \cdot \csc^2 45^\circ + 2(\sin 60^\circ + \sin 30^\circ) = \tan 60^\circ\)

8. If \(\tan 2\theta \cdot \tan 3\theta = 1\), then find the value of \(\theta\), given that \(0 \leq \theta \leq \cfrac{\pi}{2}\).

9. If \[ \tan \theta \cdot \cos 60^\circ = \frac{\sqrt{3}}{2} \] then find the value of \[ \sin(\theta - 15^\circ) \] given that \(0^\circ < \theta < 90^\circ\).

10. Find the value of \(x\):
\(x \sin 45° \cdot \cos 45° \cdot \tan 60° = \tan^2 45° - \cos^2 60°\)

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

11. Find the value of \( \tan42° \cdot \tan48° \).

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

12. 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°

13. Find the simplest value of \(\tan\cfrac{3\pi}{20} \cdot \tan\cfrac{4\pi}{20} \cdot \tan\cfrac{5\pi}{20} \cdot \tan\cfrac{6\pi}{20} \cdot \tan\cfrac{7\pi}{20}\).

14. Find the value of: \( \cfrac{1 - \sin^2 30^\circ}{1 + \sin^2 45^\circ} \times \cfrac{\cos^2 60^\circ + \cos^2 30^\circ}{\csc^2 90^\circ - \cot^2 90^\circ} \div (\sin 60^\circ \cdot \tan 30^\circ) \)

15. If \(\tan 55^\circ \cdot \tan 35^\circ = \sin A\), then find the least positive value of \(A\).

(a) 0° (b) 35° (c) 55° (d) 90°

16. Evaluate the value of \(\cfrac{2}{3} \cosec^2 58^\circ - \cfrac{2}{3} \cot 58^\circ \cdot \tan 32^\circ - \cfrac{5}{3} \tan 13^\circ \cdot \tan 37^\circ \cdot \tan 45^\circ \cdot \tan 53^\circ \cdot \tan 77^\circ\)

17. If \( \tan \theta \cdot \tan \phi = 1 \), then the value of \( \sin\left(\frac{\theta + \phi}{3}\right) \) will be –

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

18. Evaluate the value of: \(\cfrac{\tan60^\circ - \tan30^\circ}{1 + \tan60^\circ \cdot \tan30^\circ} + \cos60^\circ \cdot \cos30^\circ + \sin60^\circ \cdot \sin30^\circ\)"

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. 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

21. 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

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

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

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

24. 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) \(-\cfrac{1}{3}\)

25. 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°

26. 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) \(\cfrac{\sqrt3}{2}\)

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

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

(a) 60° (b) 45° (c) 30° (d) 90°

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

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

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