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

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

2. Find the simplest value of \(\tan\cfrac{3\pi}{20} \cdot \tan\cfrac{4\pi}{20} \cdot \tan\cfrac{5\pi}{20} \cdot \tan\cfrac{6\pi}{20} \cdot \tan\cfrac{7\pi}{20}\).

3. If \(\tan 4\theta \times \tan 6\theta = 1\) and \(6\theta\) is a positive acute angle, find the value of \(\theta\).

4. If \(x\) is a real positive number and \(\sin x = \frac{2}{3}\), then what is the value of \(\tan x\)?

(a) \(\cfrac{2}{\sqrt5}\) (b) \(\cfrac{\sqrt5}{2}\) (c) \(\sqrt{\cfrac{5}{3}}\) (d) \(\cfrac{\sqrt5}{\sqrt2}\)

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

6. 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) \(\cfrac{2}{3}\)

7. What is the value of \( \tan 1^\circ \times \tan 2^\circ \times \tan 3^\circ \times \ldots \times \tan 89^\circ \)?

8. If \(\frac{\sinθ + \cosθ}{\sinθ - \cosθ} = 7\), then what is the value of \(\tanθ\)?

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

10. If \(\cos 52^\circ = \frac{x}{\sqrt{x^2 + y^2}}\), then what is the value of \(\tan 38^\circ\)?

11. If \(\sin(2x + y) = \cos(4x - y)\), find the value of \(\tan 3x\).

12. If \(\sin 3x = 1\), what is the value of \(\tan 2x\)?

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

14. If \(\tan \alpha = \cot \beta\), find the value of \(\cos(\alpha + \beta)\), where \(0^\circ < \alpha, \beta < 90^\circ\).

15. Given: \(\tan A = \frac{x}{y}\), find the value of \(\frac{\cos A - \sin A}{\cos A + \sin A}\).

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

17. If \(\cos 43^\circ = \frac{x}{\sqrt{x^2 + y^2}}\), then what is the value of \(\tan 47^\circ\)?

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

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

20. 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

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

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

22. 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

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

24. 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) \(1\)

25. 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) \(\cfrac{c-a}{a-b}\)

26. 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) \(\sqrt{a^3-b^3}\)

27. 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

28. Find the simplest value of \(sin^2θ + \cfrac{1}{1 + \tan^2θ}\).

(a) \(2\) (b) \(1\) (c) \(0\) (d) \(\sqrt3\)

29. If \( \tan 4θ \times \tan 6θ = 1 \) and \( 6θ \) is a positive acute angle, then find the value of \( θ \).

(a) \(5°\) (b) \(10°\) (c) \(9°\) (d) \(4°\)

30. If \( \tan 4\theta \times \tan 6\theta = 1 \) and \( 6\theta \) is an acute positive angle, find the value of \( \theta \).