1. If \(3x = \csc \alpha\) and \(y = \cot \alpha\), then the value of \(3\left(x^2 - \cfrac{1}{x^2}\right)\) is -

2. If \(3x = \csc\alpha\) and \(\frac{3}{x} = \cot\alpha\), find the value of \(\left(x^2 - \frac{1}{x^2}\right)\).

3. If \(\alpha\) and \(\beta\) are the two roots of the quadratic equation \(3x^2 + 2x - 5 = 0\), then find the value of \(\cfrac{\alpha^2}{\beta} + \cfrac{\beta^2}{\alpha}\).

4. If \( 3x = \csc \alpha \) and \( \cfrac{3}{x} = \cot \alpha \), then find the value of \( 3\left(x^2-\cfrac{1}{x^2}\right) \).

(a) \(\cfrac{1}{27}\) (b) \(\cfrac{1}{81}\) (c) \(\cfrac{1}{3}\) (d) \(\cfrac{1}{9}\)

5. If \(\csc θ + \cot θ = \sqrt{3}\), then find the values of both \(\csc θ\) and \(\cot θ\).

6. If \( 3x = \csc α \) and \( \cfrac{3}{x} = \cot α \), then find the value of \( 3\left(x^2 - \cfrac{1}{x^2}\right) \).

(a) \(\cfrac{1}{27}\) (b) \(\cfrac{1}{81}\) (c) \(\cfrac{1}{3}\) (d) \(\cfrac{1}{9}\)

7. If \( \csc^2 θ = 2\cot θ \) and \( 0° < θ < 90° \), then find the value of \( θ \).

8. If \(\cot A = \frac{4}{7.5}\), then find the values of \(\cos A\) and \(\csc A\), and show that: \[ 1 + \cot^2 A = \csc^2 A \]

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

10. If \( \csc^2 θ = 2 \cot θ \), then find the value of \( θ \), where \( 0^\circ < θ < 90^\circ \).

11. If \( \tan 2A = \cot (A - 18^\circ) \) and \(2A\) is a positive acute angle, then find the value of \(A\).

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

13. If \(\alpha\) and \(\beta\) are the roots of the equation \(x^2 - 3x + 5 = 0\), then find the value of \((\alpha + \beta)\left(\frac{1}{\alpha^2} + \frac{1}{\beta^2}\right)\).

14. If \( \cosθ = \cfrac{3}{5} \), then find the value of \( \cotθ + \cscθ \).

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

15. If \( \csc A = \sqrt2 \), then find the value of
\(\cfrac{2\sin^2A + 3\cot^2A}{4\tan^2A - \cos^2A}\).

(a) \(\cfrac{8}{7}\) (b) \(\cfrac{7}{8}\) (c) \(\cfrac{1}{8}\) (d) \(\cfrac{1}{7}\)

16. If \(\alpha\) and \(\beta\) are the roots of the equation \(3x^2+8x+2=0\), find the value of \(\cfrac{1}{\alpha^2}+\cfrac{1}{\beta^2}\).

17. If the quadratic equation \(x^2 + px + q = 0\) has roots \(\alpha\) and \(\beta\), then find the value of \(\alpha^3 + \beta^3\).

18. 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

19. If \(\sin θ = \frac{4}{5}\), then find the value of \(\frac{\csc θ}{1 + \cot θ}\).

20. If \(\tan^2 θ + \cot^2 θ = \frac{10}{3}\), then find the values of \(\tan θ + \cot θ\) and \(\tan θ - \cot θ\), and from there calculate the value of \(\tan θ\).

21. If \( \sinθ – \cosθ = 0 \) (where \( 0° ≤ θ ≤ 90° \)) and \( \secθ + \cscθ = x \), then find the value of \( x \).

(a) 1 (b) 2 (c) \(\sqrt2\) (d) \(2\sqrt2\)

22. If \(\cot θ = 2\), then find the values of \(\tan θ\) and \(\sec θ\), and show that: \[1 + \tan^2θ = \sec^2θ\]

23. If \(\sin(A + B) = 1\) and \(\cos(A - B) = 1\), then find the value of \(\cot 2A\), given that \(0^\circ \leq (A + B) \leq 90^\circ\) and \(A \geq B\).

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

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 \( \tan\theta \).