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

2. If \(0^\circ < \theta < 90^\circ\), then show that \(\sin\theta + \cos\theta > 1\)

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

4. If \(r \cos \theta = \frac{1}{2}\) and \(r \sin \theta = \frac{\sqrt{3}}{2}\), then find the value of \(r\), where \(0^\circ < \theta < 90^\circ\).

5. If \(a(\tan\theta + \cot\theta) = 1\) and \(\sin\theta + \cos\theta = b\), then prove that \(2a = b^2 - 1\), where \(0^\circ < \theta < 90^\circ\).

6. If \(\sin \theta = \cfrac{p^2 - q^2}{p^2 + q^2}\), then show that \(\cot \theta = \cfrac{2pq}{p^2 - q^2}\) where \(p > q\) and \(0^\circ < \theta < 90^\circ\).

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

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

9. If \(0^\circ < \theta < 90^\circ\), then \(\sin\theta < \sin^2\theta\).

10. If \( r \cos\theta = 2\sqrt{3} \), \( r \sin\theta = 2 \), and \( 0^\circ < \theta < 90^\circ \), then find the values of \( r \) and \( \theta \).

11. If \( r \cos\theta = 2\sqrt3 \), \( r \sin\theta = 2 \), and \( 0^\circ < \theta < 90^\circ \), then determine the values of \( r \) and \( \theta \).

12. If \(0^\circ \le \theta \le 90^\circ\) and \(3 - 3\sin\theta - \cos^2\theta = 0\), then find the value of \(\theta\).

(a) \(30^o\) (b) \(60^o\) (c) \(90^o\) (d) \(45^o\)

13. If \(\sec \theta = \csc \phi\), where \(0^\circ < \theta < 90^\circ\) and \(0^\circ < \phi < 90^\circ\), then the value of \(\sin(\theta + \phi)\) is 1.

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 \(0^\circ \leq \alpha < 90^\circ\), find the minimum value of \((\sec^2α + \cos^2α)\).

(a) 1 (b) 2 (c) \(\cfrac{5}{2}\) (d) 0

16. If \(0^\circ < θ < 90^\circ\), then \( \sin θ > \sin^2 θ \).

17. If \( \csc \theta + \cot \theta = \sqrt{3} \), then find the value of \( \sin \theta \), where \( 0^\circ < \theta < 90^\circ \).

18. 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) \(\cfrac{1}{7}\)

19. Determine the values of \(\theta\) for which \(\sin^2\theta - 3\sin\theta + 2 = 0\) holds true, given that \(0^\circ < \theta < 90^\circ\).

20. Let the smaller pillar be AB \(= x\) meters and the larger pillar be CD \(= 2x\) meters.
The midpoint of the base connection BD is O; the angles of elevation from O to the tops of the pillars are \(\angle\)AOB = \(\theta\) and \(\angle\)COD = 90\(^o-\theta\).
Since O is the midpoint of BD, we have BO = OD = \(\frac{120}{2}\) meters = 60 meters.

From \( \triangle \)ABO, we get:
\(\cfrac{AB}{BO} = \tan \theta\)
Or, \(\cfrac{x}{60} = \tan\theta\) --------(i)

From \( \triangle \)COD, we get:
\(\cfrac{CD}{OD} = \tan(90^o - \theta)\)
Or, \(\cfrac{2x}{60} = \cot\theta\) --------(ii)

Multiplying equations (i) and (ii), we get:
\(\cfrac{x}{60} \times \cfrac{2x}{60} = \tan\theta \times \cot\theta\)
Or, \(\cfrac{2x^2}{60 \times 60} = 1\)
Or, \(x^2 = \cfrac{60 \times \cancel{60}30}{\cancel{2}}\)
Or, \(x = 30\sqrt2\)

\(\therefore\) The height of the smaller pillar is \(30\sqrt2\) meters.
And the height of the larger pillar is \(30\sqrt2 \times 2\) meters \(= 60\sqrt2\) meters.
(Proved).

21. Given \( r\cos\theta = 2\sqrt{3} \) and \( r\sin\theta = 2 \), where \( 0^\circ < \theta < 90^\circ \), find the values of \( r \) and \( \theta \).

22. If \(r \cos θ = 2\sqrt{3}\), \(r \sin θ = 2\), and \(0^\circ < θ < 90^\circ\), then find the values of \(r\) and \(θ\).

23. 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

24. If \( \cos^2\theta - \sin^2\theta = \frac{1}{2} \), then what is the value of \( \tan\theta \)?

(a) \(-\cfrac{1}{\sqrt3}\) (b) \(\cfrac{1}{3}\) (c) \(\cfrac{1}{\sqrt3}\) (d) \(-\cfrac{1}{3}\)

25. If \( \cos^2\theta – \sin^2\theta = \frac{1}{2} \), then what is the value of \( \tan\theta \)?

(a) \(\frac{1}{\sqrt3}\) (b) \(\sqrt3\) (c) 1 (d) None of the above

26. If \(\cos^2\theta - \sin^2\theta = \frac{1}{2}\), then the value of \(\tan\theta\) is —

(a) \(-\cfrac{1}{\sqrt3} (b) \(\cfrac{1}{3}\) (c) \(\cfrac{1}{\sqrt3}\) (d) \(\cfrac{2}{3}\)

27. If \( x \cos\theta = 3 \) and \( y \cot\theta = 4 \), then find the relationship between \( x \) and \( y \) eliminating \( \theta \).

28. If \(\cos\theta = \frac{x}{\sqrt{x^2 + y^2}}\), then prove that \(x \sin \theta = y \cos \theta\).

29. If \(x = a\cos(90^\circ - \theta)\), \(y = b\cot(90^\circ - \theta)\), then prove that \(\cfrac{a^2}{x^2} - \cfrac{b^2}{y^2} = 1\).

30. If \( \frac{\sin\theta + \cos\theta}{\sin\theta - \cos\theta} = 5 \), then find the value of \( \tan\theta \).