1. 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) \(\cfrac{x^2-y^2}{x^2+y^2}\)

2. 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) \(\cfrac{y^2-x^2}{x^2+y^2}\) (c) \(\cfrac{x^2+y^2}{y^2-x^2}\) (d) None of the above

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

4. If \(\frac{\sin \theta + \cos \theta}{\sin \theta - \cos \theta} = 5\), then what is the value of \(\tan \theta\)?

5. If \(\tan\theta = \frac{8}{15}\), then what is the value of \(\sqrt{\frac{1 - \sin\theta}{1 + \sin\theta}}\)?

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

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. If \(5\cos\theta + 12\sin\theta = 13\), then what is the value of \(\tan\theta\)?

(a) \(\cfrac{13}{15}\) (b) \(\cfrac{12}{5}\) (c) \(\cfrac{5}{13}\) (d) \(\cfrac{5}{12}\)

8. If \(\tan(\theta + 15^\circ) = \sqrt{3}\), then what is the value of \(\sin \theta\)?

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

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

11. If \(\tan(\theta + 15^\circ) = 1\), then what is the value of \(\cos 2\theta\)?

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

12. If \(a \sin^2 \theta + b \cos^2 \theta = c\), then what is the value of \(\tan^2 \theta\)?

(a) \(\cfrac{b+c}{a+c}\) (b) \(\cfrac{b-c}{c-a}\) (c) \(\cfrac{a-b}{b-c}\) (d) \(\cfrac{c-a}{a-b}\)

13. If \(\tan\theta = 3\), then what is the value of \(a\sin\theta + b\cos\theta\)?

(a) \(\sqrt{a^2+b^2}\) (b) \(\sqrt{a^2-b^2}\) (c) \(\sqrt{a^3+b^3}\) (d) \(\sqrt{a^3-b^3}\)

14. If \( \tan\theta = \cfrac{4}{3} \), then what is the value of \( \sin\theta + \cos\theta \)?

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

16. 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

17. What is the value of \(\cfrac{1}{\tan\theta+\cfrac{1}{\tan\theta}}\)?

(a) \(tan\theta\) (b) \(sin 2\theta\) (c) \(cos \theta\) (d) \(sin\theta cos\theta\)

18. If \(\theta\) is a positive acute angle and \(\sin \theta = \frac{\sqrt{3}}{2}\), then what is the value of \(\tan(\theta - 15^\circ)\)?

19. 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

20. What is the value of \(\cfrac{\sin \theta}{\csc \theta} + \cfrac{\cos \theta}{\sec \theta} - 2\)?

(a) 0 (b) -1 (c) 4 (d) None of the above

21. If \( \cos\theta - \sin\theta = \sqrt{2} \sin\theta \), then what is the value of \( \cos\theta + \sin\theta \)?

(a) \(2 cos\theta\) (b) \(\sqrt2 sin \theta\) (c) \(2 sin\theta\) (d) \(\sqrt2 cos \theta\)

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

23. If \(\csc\theta + \cot\theta = 2 + \sqrt{3}\), then what is the value of \(\csc\theta - \cot\theta\)?

(a) \(\sqrt3+\sqrt2\) (b) \(\sqrt2-3\) (c) \(\sqrt3-2\) (d) \(2-\sqrt3\)

24. If \(x = r\cos\theta\cos\phi\), \(y = r\cos\theta\sin\phi\), and \(z = r\sin\theta\), then what is the value of \(x^2 + y^2 + z^2\)?

(a) \(r\) (b) \(1\) (c) \(r^2\) (d) \(-r^2\)

25. 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) \(\cfrac{10}{13}\)

26. If \( \tan\theta + \cot\theta = 2 \), then the value of \( \tan\theta - \cot\theta \) is —

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

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

(a) 1 (b) 3 (c) 2 (d) 4

28. If \( \csc\theta \tan\theta = 2 \), then what is the value of \( \theta \)?

(a) 60° (b) 45° (c) 30° (d) 90°

29. If \( \sin\theta + \csc\theta = 2 \), then what is the value of \( \sin^2\theta + \csc^2\theta \)?

(a) 1 (b) 2 (c) 4 (d) \cfrac{3}{4}

30. 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