1. Show that \[ \frac{2 \tan^2 30^\circ}{1 - \tan^2 30^\circ} + \sec^2 45^\circ - \cot^2 45^\circ = \sec 60^\circ \]

2. Show that: \[ \cfrac{2 \tan^2 30^\circ}{1 - \tan^2 30^\circ} + \sec^2 45^\circ - \cot^2 45^\circ = \sec 60^\circ \]

3. If \[ x \cos 60^\circ = \frac{2 \tan 45^\circ}{1 + \tan^2 45^\circ} - \frac{1 - \tan^2 30^\circ}{1 + \tan^2 30^\circ} \] then find the value of \(x\).

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

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

6. Evaluate the expression: \[ \cot^2 30^\circ - 2\cos^2 60^\circ - 4\sin^2 30^\circ - \frac{3}{4}\sec^2 45^\circ + \tan 45^\circ \]

7. Evaluate the expression: \[ \frac{1 - \sin^2 30^\circ}{1 + \sin^2 45^\circ} \times \frac{\cos^2 60^\circ + \cos^2 30^\circ}{\csc^2 90^\circ - \cot^2 90^\circ} \div (\sin 60^\circ \tan 30^\circ) \]

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

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

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

11. Evaluate the following expression: \[ \sec^2 60^\circ - \cot^2 30^\circ - \frac{2 \tan 30^\circ \csc 60^\circ}{1 + \tan^2 30^\circ} \]

12. If \(x^2 = \sin^2 30^\circ + 4\cot^2 45^\circ - \sec^2 60^\circ\), then the value of \(x\) is —

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

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

14. Evaluate: \[ 3\tan^2 45^\circ - \sin^2 60^\circ - \frac{1}{3} \cot^2 30^\circ - \frac{1}{8} \sec^2 45^\circ \]

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

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

17. If \(x = \sin^2 30^\circ + 4 \cot^2 45^\circ - \sec^2 60^\circ\), find the value of \(x\).

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

19. Solve: \[ x^2 = \sin^2 30^\circ + 4\cot^2 45^\circ - \sec^2 60^\circ \]

20. If \(x^2 = \sin^2 30^\circ + 4\cot^2 45^\circ - \sec^2 60^\circ\), then what is the value of \(x\)?

21. 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

22. If \(x^2 = \sin^2 30^\circ + 4\cot^2 45^\circ - \sec^2 60^\circ\), find the value of \(x\). Let me know if you'd like a step-by-step solution next. I'm ready when you are.

23. If \(x^2 = \sin^2 30^\circ + 4\cot^2 45^\circ - \sec^2 60^\circ\), find the value of \(x\).

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

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

26. If \(x^2 = \sin^2 30^\circ + 4\cot^2 45^\circ - \sec^2 60^\circ\), then find the value of \(x\).

27. If \(x^2 = \sin^2 30^\circ + 4\cot^2 45^\circ - \sec^2 60^\circ\), then find the value of \(x\).