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

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

3. If \(\cfrac{\sin \theta + \cos \theta}{\sin \theta - \cos \theta} = 7\), then find the value of \(\tan \theta\).

4. If \( \tan\theta - \cot\theta = 0 \), then find the value of \( \sin(\theta - 15^\circ) + \cos(\theta + 15^\circ) \).

5. If \( \tan\theta = \cfrac{4}{3} \), then find the value of \( \sin\theta + \cos\theta \).

6. If \( \tan \theta \cos 60° = \cfrac{√3}{2} \), find the value of \(\sin(\theta - 15°)\)

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

7. In a right-angled triangle, the two acute angles are \(\theta\) and \(\phi\). If \( \tan\theta = \cfrac{5}{12} \), then what is the value of \( \sin\phi \)?

(a) \(\cfrac{12}{13}\) (b) \(\cfrac{5}{13}\) (c) \(\cfrac{1}{4}\) (d) \(\cfrac{10}{13}\)

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

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

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

11. If \(\tan 2\theta \cdot \tan 3\theta = 1\), then find the value of \(\theta\), given that \(0 \leq \theta \leq \cfrac{\pi}{2}\).

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

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

14. If \(\tan \theta = \cfrac{8}{15}\), then the value of \(\sqrt{\cfrac{1 - \sin \theta}{1 + \sin \theta}}\) is —

(a) \(\cfrac{2}{5}\) (b) \(\cfrac{3}{5}\) (c) \(\cfrac{1}{5}\) (d) None of the above

15. If \( \tan\theta = \frac{x}{y} \), then find the value of \[ \frac{x \sin\theta - y \cos\theta}{x \sin\theta + y \cos\theta} \]

16. If \[ \frac{\sec\theta + \tan\theta}{\sec\theta - \tan\theta} = 2\frac{51}{79} \] then find the value of \(\sin\theta\).

17. If \(\sec \theta - \tan \theta = \frac{1}{2}\), then find the values of \(\sec \theta\) and \(\tan \theta\).

18. If \( \cos\theta = \frac{p}{\sqrt{p^2 + q^2}} \), then find the value of \( \tan\theta \).

(a) \(\cfrac{q}{p}\) (b) \(\cfrac{p}{q}\) (c) \(\cfrac{\sqrt{p}}{q}\) (d) None of the above

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

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

21. If \( \sin \theta - \cos \theta = 0 \) (where \( 0^\circ \lt \theta \lt 90^\circ \)) and \( \sec \theta + \csc \theta = x \), then find the value of \( x \).

(a) 1 (b) 2 (c) √2 (d) 2√2

22. Given that \(\sin 10\theta = \cos 8\theta\) and \(10\theta\) is a positive acute angle, find the value of \(\tan 9\theta\).

23. 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) \(45^o\)

24. If \( \cos^4\theta - \sin^4\theta = \frac{2}{3} \), find the value of \( 1 - 2\sin^2\theta \).

25. 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 \( \tan\theta \).

26. If \( \sin\theta + \csc\theta = 2 \), then find the value of \( \sin^{10}\theta + \csc^{10}\theta \).

27. If \(\cos^4\theta - \sin^4\theta = \cfrac{2}{3}\), then find the value of \(1 - 2\sin^2\theta\).

28. If \( \tan \theta = \cfrac{5}{7} \), then find the value of \[ \cfrac{5\sin \theta + 7\cos \theta}{7\sin \theta + 5\cos \theta}. \]

29. 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) \(-\cfrac{1}{3}\)

30. If \( \tan \theta = \frac{x}{y} \), then what is the value of \[ \frac{x\sin\theta - y\cos\theta}{x\sin\theta + y\cos\theta}? \]

(a) \(\cfrac{x^2+y^2}{x^2-y^2}\) (b) \(\cfrac{x-y}{x+y}\) (c) \(\cfrac{x+y}{x-y}\) (d) \(\cfrac{x^2-y^2}{x^2+y^2}\)