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

(a) \(0\) (b) \(1\) (c) \(\sqrt2\) (d) None of the above

2. If \(\cot \theta = \cfrac{x}{y}\), then the value of \(\cfrac{x\cos\theta - y\sin\theta}{x\cos\theta + y\sin\theta}\) is —

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

3. If \( r\cos\theta = 1 \) and \( r\sin\theta = \sqrt{3} \), then the value of \( \theta \) is—

(a) \(\cfrac{π}{2}\) (b) \(\cfrac{π}{3}\) (c) \(\cfrac{π}{4}\) (d) \(\cfrac{π}{6}\)

4. 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) \(\sqrt2 cos \theta\)

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

6. 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) \(-r^2\)

7. 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

8. 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) \(\cfrac{5}{12}\)

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

(a) \(\sqrt2 sin\theta\) (b) \(-\sqrt2 sin\theta\) (c) \(-\sqrt2 cos\theta\) (d) None of the above

10. 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

11. If \( a\cos\theta + b\sin\theta = c \), then what is the value of \( a\sin\theta - b\cos\theta \)?

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

12. 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

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 \(\sin\theta + \cos\theta = \sqrt{2}\), then the value of \(\theta\) will be—

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

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

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

(a) 0 (b) 1 (c) 2 (d) None of the above

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

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

19. If \( \sin\theta = \cos\theta \), then the value of \( 2\theta \) is –

(a) 90° (b) 60° (c) 45° (d) 30°

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

21. If \( r \sin\theta = \frac{7}{2} \) and \( r \cos\theta = \frac{7\sqrt{3}}{2} \), then what are the values of \( r \) and \( \theta \)?

(a) \(r=7, \theta=30^0\) (b) \(r=\cfrac{1}{7}, \theta=30^0\) (c) \(r=7, \theta=60^0\) (d) \(r=\cfrac{1}{7}, \theta=60^0\)

22. 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}

23. The simplest value of \(\sin\theta \times \tan\theta + \cos\theta\) is —

(a) \(cos\theta\) (b) \(tan\theta\) (c) \(cosec\theta\) (d) \(sec\theta\)

24. If \(\theta\) is a positive acute angle and \( \sin\theta = \cos(2\theta + 15^\circ) \), then what is the value of \(\theta\)?

(a) 30° (b) 25° (c) 60° (d) 90°

25. If \(\cos\theta = \frac{4}{5}\), then what is the value of \(\csc\theta \div (1 + \cot\theta)\)?

(a) \(\cfrac{2}{4}\) (b) \(\cfrac{5}{7}\) (c) \(\cfrac{3}{5}\) (d) \(\cfrac{1}{3}\)

26. 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

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

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

29. If \(\cos\theta + \sec\theta = 2\), then what is the value of \(\cos^{11}\theta + \sec^{11}\theta\)?

(a) 0 (b) 1 (c) 2 (d) 22

30. If \( \sin\theta = \frac{1}{2} \), then the value of \( \theta \) is —

(a) 30° (b) 60° (c) 45° (d) 15°