1. Prove that, \[ \cot 12^\circ \cdot \cot 38^\circ \cdot \cot 52^\circ \cdot \cot 78^\circ \cdot \cot 60^\circ = \frac{1}{\sqrt{3}} \]

2. If \(α\) and \(β\) are two complementary angles, show that \[ \sqrt{\frac{\tan α}{\cot β} + \tan α \cdot \cot β} = \sec α \]

3. Left-hand side (LHS): \[ 1 + \cfrac{\tan A}{\tan B} = 1 + \cfrac{\tan(90^\circ - B)}{\tan B} = 1 + \cfrac{\cot B}{\cot B} = 1 + \cot^2 B = \csc^2 B \] Right-hand side (RHS): \[ \tan^2 A \cdot \sec^2 B = \tan^2(90^\circ - B) \cdot \sec^2 B = \cot^2 B \cdot \sec^2 B = \cfrac{\cos^2 B}{\sin^2 B} \cdot \cfrac{1}{\cos^2 B} = \cfrac{1}{\sin^2 B} = \csc^2 B \] \(\therefore\) LHS = RHS (Proved)

4. Show that: \(\csc^2 22^\circ \cdot \cot^2 68^\circ = \sin^2 22^\circ + \sin^2 68^\circ + \cot^2 68^\circ\)

5. Prove that \[ \frac{\tan 47^\circ + \cot 27^\circ}{\tan 43^\circ + \cot 63^\circ} = \tan 47^\circ \cdot \cot 27^\circ \]

6. Show that, \[ \csc^2 22^\circ \cdot \cot^2 68^\circ = \sin^2 22^\circ + \sin^2 68^\circ + \cot^2 68^\circ \]

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

8. Prove that: \( \csc^2 22^\circ \cdot \cot^2 68^\circ = \sin^2 22^\circ + \sin^2 68^\circ + \cot^2 68^\circ \)

9. Prove: \[ \csc^2 22^\circ \cdot \cot^2 68^\circ = \sin^2 22^\circ + \sin^2 68^\circ + \cot^2 68^\circ \]

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

11. If \( \tan 4\theta \cdot \tan 6\theta = 1 \), then \( \tan 5\theta = \cot 5\theta = 1 \) ∴ \( \tan 5\theta = 1 \)

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

13. 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

14. Simplify: \( \cfrac{\sin\left(\cfrac{\pi}{2}-θ\right) \cdot \cot\left(\cfrac{\pi}{2}-θ\right)}{\sinθ} \).

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

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

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