1. 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
2. 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) \(-\cfrac{1}{3}\)
3. 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
4. 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) \(\cfrac{5}{12}\)
5. 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
6. If \( \sin\theta \cos\theta = \frac{1}{2} \), then what is the value of \( (\sin\theta + \cos\theta)^2 \)?
7. If \(\cos^4\theta - \sin^4\theta = \frac{2}{3}\), then what is the value of \(1 - 2\sin^2\theta\)?
8. 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
9. 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) \(1\)
10. 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°
11. 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
12. If \(2\cos(3\theta) = 1\), then what is the value of \(\theta\)?
(a) \(10^\circ\) (b) \(15^\circ\) (c) \(20^\circ\) (d) \(30^\circ\)
13. If \(\sin 3\theta = \frac{1}{2}\), then what is the value of \(\cos(80^\circ + \theta)\)?
14. If \(\sin (\theta + 30^\circ) = \cos 15^\circ\), then what is the value of \(\cos 2\theta\)?
15. 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) \(\sqrt2 cos \theta\)
16. 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) \(-r^2\)
17. 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) \(\cfrac{10}{13}\)
18. 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
19. If \(\sin \theta + \cos \theta = \sqrt{2}\), then what is the value of \(\theta\)?
(a) \(\cfrac{\pi}{2}\) (b) \(\cfrac{\pi}{3}\) (c) \(\pi\) (d) \(\cfrac{\pi}{4}\)
20. If \(\sec^2 \theta + \tan^2 \theta = \frac{13}{12}\), then what is the value of \(\sec^4 \theta - \tan^4 \theta\)?
21. If \(x = 3 \cos \theta\) and \(y = 3 \sin \theta\), then what is the value of \(x^2 + y^2\)?
22. If \(\theta\) is a positive acute angle and \(\sin \theta = \frac{\sqrt{3}}{2}\), then what is the value of \(\tan(\theta - 15^\circ)\)?
23. 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) \(\cfrac{c-a}{a-b}\)
24. 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) \(\cfrac{1}{3}\)
25. If \(2\sin 2\theta - \sqrt{3} = 0\), then what is the value of \(\csc \theta\)?
(a) \(\cfrac{1}{2}\) (b) 1 (c) \(\cfrac{2}{\sqrt3}\) (d) 2
26. 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
27. 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) \(r=\cfrac{1}{7}, \theta=60^0\)
28. 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) \(\cfrac{x^2-y^2}{x^2+y^2}\)
29. 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) \(2-\sqrt3\)
30. From the equation \( \sin(90^\circ + \theta) = \cos(120^\circ - 3\theta) \), the value of \( \theta \) is —
(a) 30° (b) 60° (c) 45° (d) None of the above
