1. If \(0^\circ < \theta < 90^\circ\), then determine the minimum value of \((9 \tan^2 \theta + 4 \cot^2 \theta)\).

2. If \(0^\circ \le \theta \le 90^\circ\), then find the minimum value of \[ 9\tan^2\theta + 4\cot^2\theta \]

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

4. Find the value of \(\cfrac{4}{\sec^2\theta}+\cfrac{1}{1+\cot^2\theta}+3\sin^2\theta\).

5. If \( 0° < θ < 90° \), then find the minimum value of \( 9 \tan^2 θ + 4 \cot^2 θ \).

6. 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

7. 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

8. If \(\sec^2 \theta + \tan^2 \theta = \frac{13}{12}\), then what is the value of \(\sec^4 \theta - \tan^4 \theta\)?

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

10. If \( \csc A = \sqrt2 \), then find the value of
\(\cfrac{2\sin^2A + 3\cot^2A}{4\tan^2A - \cos^2A}\).

(a) \(\cfrac{8}{7}\) (b) \(\cfrac{7}{8}\) (c) \(\cfrac{1}{8}\) (d) \(\cfrac{1}{7}\)

11. What is the value of \[ \left(\cfrac{4}{\sec^2 \theta}+\cfrac{1}{1+\cot^2 \theta}+3 \sin^2 \theta \right) \]

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

13. Find the numerical value of \(\frac{9}{\csc^2\theta} + 4\cos^2\theta + \frac{5}{1 + \tan^2\theta}\).

14. If \(0^\circ < \theta \leq 90^\circ\), then what is the minimum value of \((4\csc^2\theta + 9\sin^2\theta)\)?

15. If \( \tan^2\theta + \cot^2\theta = \cfrac{10}{3} \), then find the values of \( \tan\theta + \cot\theta \) and \( \tan\theta - \cot\theta \). From there, determine the value of \( \tan\theta \).

16. What is the value of \(\frac{4}{\sec^2\theta} + \frac{1}{1 + \cot^2\theta} + 3\sin^2\theta\)?

17. Find the value of > \[ \left(\frac{4}{\sec^2\theta} + \frac{1}{1 + \cot^2\theta} + 3\sin^2\theta\right) \]

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

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

19. The minimum and maximum values of \(a + b\sin\theta\) are 5 and 6 respectively. Find the values of \(a\) and \(b\).

(a) \(a=1, b=5\) (b) \(a=-5, b=1\) (c) \(a=5, b=1\) (d) \(a=-1, b=5\)

20. 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) \(\pm \cfrac{1}{\sqrt3}\)

21. The minimum value of \( \cos^2\theta + \sec^2\theta \) is:

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

22. What is the value of \[ \frac{4}{\sec^2θ} + \frac{1}{1 + \cot^2θ} + 3\sin^2θ? \]

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

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

25. If \(\alpha\) and \(\beta\) are complementary angles, find the value of the expression: \[ (1 - \sin^2 \alpha)(1 - \cos^2 \alpha)(1 + \cot^2 \beta)(1 + \tan^2 \beta) \]

26. From the equation \(5 \sin^2 \theta + 4 \cos^2 \theta = \frac{9}{2}\), find the value of \(\tan \theta\), where \(0^\circ < \theta < 90^\circ\).

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

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

29. If \(\sin 4\theta = \cos 5\theta\), find the value of \(\theta\).

30. If \(2 \cos^2\theta + 3 \sin \theta = 3\) and \(0^\circ < \theta < 90^\circ\), find the value of \(\theta\).