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

2. If \(3\sinθ + 4\cosθ = 5\), then what is the value of \(4\sinθ - 3\cosθ\)?

3. If \(\tan θ + \cot θ = 2\), then find the value of \(\tan θ - \cot θ\).

4. 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 \(\tan θ\).

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

6. If \( \tanθ + \cotθ = 2 \), the value of \( \tanθ - \cotθ \) will be

(a) 2 (b) 0 (c) -2 (d) \(\cfrac{1}{2}\)

7. If \( \cos^2θ − \sin^2θ = \frac{1}{x} \), where \( x > 1 \), then what is the value of \( \cos^4θ − \sin^4θ \)?

8. If \( \cotθ = \cfrac{15}{8} \), then find the value of
\(\cfrac{(2+2\sinθ)(1-\sinθ)}{(1+\cosθ)(2-2\cosθ)}\).

(a) \(0\) (b) \(225\) (c) \(64\) (d) \(\cfrac{225}{64}\)

9. If \( \sinθ = \cfrac{a}{\sqrt{a^2+b^2}}; 0° < θ < 90° \), then find the value of \( \tanθ \).

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

10. If \( \tanθ = \cfrac{1}{\sqrt7} \), then find the value of
\(\cfrac{\csc^2θ - \sec^2θ}{\csc^2θ + \sec^2θ}\).

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

11. If \( \sinθ + \cosθ = \sqrt2 \sin(90° – θ) \), then find the value of \( \cotθ \).

(a) \(\cfrac{\sqrt2}{3}\) (b) \(1\) (c) \(\sqrt2\) (d) \(\sqrt2+1\)

12. If \(\tan θ = 1\), then find the value of \(\frac{8 \sin θ + 5 \cos θ}{\sin^3 θ - 2 \cos^3 θ + 7 \cos θ}\).

13. If \(\sec θ + \tan θ = 2\), then find the value of \(\sec θ - \tan θ\).

14. If \(\csc θ - \cot θ = \sqrt{2} - 1\), then calculate the value of \(\csc θ + \cot θ\).

15. If \(5\sin^2 θ + 4\cos^2 θ = \frac{9}{2}\), then find the value of \(\tan θ\).

16. If \(\sec^2 θ + \tan^2 θ = \frac{13}{12}\), then calculate the value of \(\sec^4 θ - \tan^4 θ\).

17. If \( 0° < θ < 90° \), then find the minimum value of \( 9 \tan^2 θ + 4 \cot^2 θ \).

18. If \(sinθ−cosθ=0,\) \( (0°<θ<90°)\) and \(secθ+cosecθ=x\), then the value of \(x\) is—?

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

19. 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

20. If \( \sin^2A + \sin^4A = 1 \), then what is the value of \( \tan^2A - \tan^4A \)?

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

21. 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) \(2-\sqrt3\)

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

23. If \( \sin 51^\circ = \cfrac{a}{\sqrt{a^2 + b^2}} \), then what is the value of \( \tan 51^\circ + \tan 39^\circ \)?

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

24. If \( \tan\alpha + \cot\alpha = \sqrt{3} \), then what is the value of \( \tan^3\alpha + \cot^3\alpha \)?

25. If \( \cot\alpha = \tan(\beta + \gamma) \), then what is the value of \( \sin(\alpha + \beta + \gamma) \)?

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

26. If \( \sqrt{2} \sin(2x + 5^\circ) = \cot 45^\circ \), then what is the value of \( \sec 3x \)?

27. If \( \tan(θ + 15^\circ) = \sqrt{3} \), then find the value of \( \sinθ + \cosθ \).

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

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

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