1. Find the value of: \( \sec^2 45^\circ - \cot^2 45^\circ - \sin^2 30^\circ - \sin^2 60^\circ \)

2. Evaluate the value of: \[ \frac{5 \cos^2 \left( \frac{\pi}{3} \right) + 4 \sec^2 \left( \frac{\pi}{6} \right) - \tan^2 \left( \frac{\pi}{6} \right)}{\sin^2 \left( \frac{\pi}{6} \right) + \cos^2 \left( \frac{\pi}{6} \right)} \]

3. Find the simplest value of: \(\sqrt7(\sqrt5 − \sqrt2) − \sqrt5(\sqrt7 − \sqrt2) + \cfrac{2\sqrt2}{\sqrt5 + \sqrt7}\)

4. Find the value of: \(\cfrac{\sec 17^\circ}{\csc 73^\circ} + \cfrac{\tan 68^\circ}{\cot 22^\circ} + \cos^2 44^\circ + \cos^2 46^\circ\)

5. Evaluate the value of: \[ \cfrac{5\cos^2\left(\cfrac{\pi}{3}\right) + 4\sec^2\left(\cfrac{\pi}{6}\right) - \tan^2\left(\cfrac{\pi}{4}\right)}{\sin^2\left(\cfrac{\pi}{6}\right) + \cos^2\left(\cfrac{\pi}{6}\right)} \]

6. Evaluate the value of: \[ \sin^2 \left(\frac{\pi}{3}\right) - \sec^2 \left(\frac{\pi}{4}\right) - \csc^2 \left(\frac{\pi}{4}\right) + \cot^2 \left(\frac{\pi}{4}\right) \]

7. Find the value of: \[ \frac{2\tan^2 30^\circ}{1 - \tan^2 30^\circ} + \sec^2 45^\circ - \cot^2 45^\circ - \sec 60^\circ \]

8. Given: \[ x = \frac{\sqrt{7} + \sqrt{3}}{\sqrt{7} - \sqrt{3}} \quad \text{and} \quad xy = 1 \] Find the value of: \[ \frac{x^2 + 3xy + y^2}{x^2 - 3xy + y^2} \]

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. Evaluate the value of:
\(sec^260° – cot^230° – \cfrac{2tan30° cosec60°}{1+tan^230°}\)

(a) \(1\) (b) \(2\) (c) \(0\) (d) \(\sqrt3\)

12. Determine the value of: \[ \cfrac{4}{1+\tan^2\theta}+\cfrac{3}{1+\cot^2\theta}+\sin^2\theta \]

13. If \(m + \cfrac{1}{m} = \sqrt{3}\), then find the simplest value of: (a) \(m^2 + \cfrac{1}{m^2}\), and (b) \(m^3 + \cfrac{1}{m^3}\).

14. Let us find the simplest value of: \(\sqrt{12}+\sqrt{18}+\sqrt{27}-\sqrt{32}\)

15. Let us evaluate the value of: \(\cfrac{2\sqrt{5}+1}{\sqrt{5}+1}-\cfrac{4\sqrt{5}-1}{\sqrt{5}-1}\)

16. Let us evaluate the value of: \(\cfrac{8+3\sqrt{2}}{3+\sqrt{5}}-\cfrac{8-3\sqrt{2}}{3-\sqrt{5}}\)

17. Find the value of: \( \sec^2 45^\circ - \cot^2 45^\circ - \sin^2 30^\circ - \sin^2 60^\circ \)

18. Find the value of: \( 3 \tan^2 45^\circ - \sin^2 60^\circ - \cfrac{1}{3} \cot^2 30^\circ - \cfrac{1}{8} \sec^2 45^\circ \)

19. Find the value of: \( \cfrac{4}{3} \cot^2 30^\circ + 3 \sin^2 60^\circ - 2 \csc^2 60^\circ - \cfrac{3}{4} \tan^2 30^\circ \)

20. Find the value of: \( \cfrac{\cfrac{1}{3} \cos 30^\circ}{\cfrac{1}{2} \sin 45^\circ} + \cfrac{\tan 60^\circ}{\cos 30^\circ} \)

21. Find the value of: \( \cot^2 30^\circ - 2 \cos^2 60^\circ - \cfrac{3}{4} \sec^2 45^\circ - 4 \sin^2 30^\circ \)

22. Find the value of: \( \sec^2 60^\circ - \cot^2 30^\circ - \cfrac{2 \tan 30^\circ \csc 60^\circ}{1 + \tan^2 30^\circ} \)

23. Find the value of: \( \cfrac{\tan 60^\circ - \tan 30^\circ}{1 + \tan 60^\circ \tan 30^\circ} + \cos 60^\circ \cos 30^\circ + \sin 60^\circ \sin 30^\circ \)

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

25. If \(x = \sqrt{3} + \sqrt{2}\) and \(y = \frac{1}{x}\), then find the value of: \[ (x + \frac{1}{x})^2 + \left( \frac{1}{y} - y \right)^2 \]

26. Find the simplest value of: \[ \cfrac{2\sqrt{7}}{\sqrt{5} + \sqrt{3}} + \cfrac{2\sqrt{3}}{\sqrt{7} + \sqrt{5}} - \cfrac{4\sqrt{5}}{\sqrt{7} + \sqrt{3}} \]

27. Evaluate the value of: \[ 3\tan^2 45^{\circ} - \sin^2 60^{\circ} - \cfrac{1}{3} \cos^2 30^{\circ} - \cfrac{1}{8} \sec^2 45^{\circ} \]

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

29. Find the minimum value of: \[ 9\csc^2 \alpha + 25\sin^2 \alpha \quad \text{when} \quad 0^\circ \leq \alpha \leq 90^\circ \]

30. Find the simplest value of: \[ \frac{3\sqrt{20} + 2\sqrt{28} + \sqrt{12}}{5\sqrt{45} + 2\sqrt{175} + \sqrt{75}} \]