1. What is the value of \(\sqrt{12+\sqrt{12+\sqrt{12+\cdots}}}\) to infinity?

(a) 3 (b) 4 (c) 12 (d) None of the above

2. What is the value of \( \tan 4^\circ \cdot \tan 43^\circ \cdot \tan 47^\circ \cdot \tan 86^\circ \)?

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

3. What is the value of \( \sin 12^\circ \cdot \cos 18^\circ \cdot \sec 78^\circ \cdot \csc 72^\circ \)?

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

4. 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

5. If \(tan 35° \cdot tan 55° = \sin θ\), then the smallest positive value of \(θ\) will be ——.

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

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

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

9. Evaluate the value of: \[ \cot\left(\frac{\pi}{20}\right) \cdot \cot\left(\frac{3\pi}{20}\right) \cdot \cot\left(\frac{5\pi}{20}\right) \cdot \cot\left(\frac{7\pi}{20}\right) \cdot \cot\left(\frac{9\pi}{20}\right) \]

(a) 3 (b) 5 (c) 1 (d) 2

10. Find the value of: \[ \cos^2 12^\circ + \cos^2 78^\circ + \tan^2 78^\circ - \sec^2 12^\circ \cdot \tan^2 78^\circ \]

(a) 0 (b) 1 (c) 2 (d) 3

11. 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

12. 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) \(\sqrt3\)

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

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

14. 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°

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

16. If \(tan\theta \cdot cos60° = \cfrac{\sqrt3}{2}\), find the value of \(sin (\theta - 15°)\).

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

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

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

20. 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°

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

22. 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) \(\cfrac{1}{2}\)

23. Determine the value of \[ \sec^2 60^\circ - \cot^2 30^\circ - \frac{2 \tan 30^\circ \cdot \csc 60^\circ}{1 + \tan^2 30^\circ} \]

24. If \[ 1\frac{1}{2} \cdot \frac{x + \sqrt{x^2 - 1}}{x - \sqrt{x^2 - 1}} + \frac{x - \sqrt{x^2 - 1}}{x + \sqrt{x^2 - 1}} = 14 \] then find the value of \(x\).

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

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

27. Determine the value of \[ \sec^2 60^\circ - \cot^2 30^\circ \cdot \cfrac{2 \tan 60^\circ \cdot \csc 60^\circ}{1 + \tan^2 30^\circ} \]

28. Evaluate the value of: \[ \sqrt{x^3\sqrt{y\sqrt{x^3\sqrt{y\sqrt{x^3\sqrt{y \cdots \infty}}}}}} \]

29. Evaluate the value of: \[ \frac{\tan{60^\circ} - \tan{30^\circ}}{1 + \tan{60^\circ} \cdot \tan{30^\circ}} + \cos{60^\circ} \cos{30^\circ} + \sin{60^\circ} \sin{30^\circ} \]

30. \[ \sum\limits_{i=1}^5 x_i = 5 \quad \text{and} \quad \sum\limits_{i=1}^5 x_i^2 = 14 \] Find the value of: \[ \sum\limits_{i=1}^5 (x_i - 3) \cdot 2x_i \]