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

2. 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

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

4. Find the value of \(\cfrac{4}{\sec^2\theta}+\cfrac{1}{1+\cot^2\theta}+3\sin^2\theta\).

5. If \( \cos\theta + \sec\theta = 2 \), find the value of \( \cos^{11}\theta + \sec^{11}\theta \).

6. 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

7. If \(\sin \theta + \sin^2 \theta = 1\), then find the value of \(\cos^2 \theta + \cos^4 \theta =\) _____

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

9. **"If \( \sec 5\theta = \csc(\theta + 36^\circ) \) and \(5\theta\) is a positive acute angle, then find the value of \( \theta \)."**

10. If \[ \frac{\sec \theta + \tan \theta}{\sec \theta - \tan \theta} = \frac{2 + \sqrt{3}}{2 - \sqrt{3}}, \] then what is the value of \( \theta \)?

(a) 60° (b) 30° (c) 45° (d) 90°

11. Find the minimum value of \(9 \tan^2 \theta + 4 \cot^2 \theta\).

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

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

15. If \[ \frac{\sin \theta + \cos \theta}{\sin \theta - \cos \theta} = 3 \] then find the value of \[ \sin^4 \theta - \cos^4 \theta \]

16. 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

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

20. 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) \(1\)

21. 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

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

23. If \(\sec \theta = \csc \phi\), where \(0^\circ < \theta < 90^\circ\) and \(0^\circ < \phi < 90^\circ\), then the value of \(\sin(\theta + \phi)\) is 1.

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

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

27. If \(\sin 3\theta = \frac{1}{2}\), then what is the value of \(\cos(80^\circ + \theta)\)?

28. If \( \csc\theta + \cot\theta = 3 \), then find the values of both \( \csc\theta \) and \( \cot\theta \).

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

30. Find the value of > \[ \left(\frac{4}{\sec^2\theta} + \frac{1}{1 + \cot^2\theta} + 3\sin^2\theta\right) \]