1. If \( \cos^2 θ - \sin^2 θ = \cfrac{1}{2} \), then find the value of \( \cos^4 θ - \sin^4 θ \).

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

3. If \( \cos^2 θ - \sin^2 θ = \cfrac{1}{2} \), then find the value of \( \tan^2 θ \).

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

5. If \(2\cos^2θ + 3\sinθ - 3 = 0\), then find the value of \(θ\).

(a) \(90°\) (b) \(30°\) (c) both (a) and (b) (d) none of the above

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

7. If \(r\cosθ = 2\sqrt{3}\), \(r\sinθ = 2\), and \(0° < θ < 90°\), then find the values of \(r\) and \(θ\).

8. If \(r\cosθ = 1\) and \(r\sinθ = \sqrt{3}\), then find the values of \(r\) and \(θ\).

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

10. If \(\cfrac{\cosθ - \sinθ}{\cosθ + \sinθ} = \cfrac{\sqrt3 - 1}{\sqrt3 + 1}\), then find the value of \(θ\).

(a) \(60°\) (b) \(90°\ (c) \(120°\) (d) \(30°\)

11. If \(x = r\sinθ \cosϕ\), \(y = r\sinθ \sinϕ\), and \(z = r\cosθ\), then find the value of \(x^2 + y^2 + z^2\).

(a) \(r\) (b) \(5r\) (c) \(\sqrt{r}\) (d) \(r^2\)

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

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

13. 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) \(\cfrac{225}{64}\)

14. 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) \(\cfrac{a^2}{b^2}\)

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

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

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

17. If \( \sinθ = \cos2θ \), then find the value of \( θ \).

(a) \(30°\) (b) \(45°\) (c) \(60°\) (d) \(90°\)

18. If \( r\cosθ = 2\sqrt{3} \), \( r\sinθ = 2 \), and \( 0^\circ < θ < 90^\circ \), then find the values of \( r \) and \( θ \).

19. If \(r \cos θ = 2\sqrt{3}\), \(r \sin θ = 2\), and \(0^\circ < θ < 90^\circ\), then find the values of \(r\) and \(θ\).

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

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

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

23. If \(\sin θ + \cos θ = 1\), then find the value of \(\sin θ × \cos θ\).

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

25. If \(\sin θ - \cos θ = \frac{7}{13}\), then find the value of \(\sin θ + \cos θ\).

26. If \(\sin θ \cos θ = \frac{1}{2}\), then calculate the value of \((\sin θ + \cos θ)\).

27. If \(\sec θ - \tan θ = \frac{1}{\sqrt{3}}\), then find the values of both \(\sec θ\) and \(\tan θ\).

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

29. 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 \(\tan θ\).

30. If \( \sinθ + \sin^2θ = 1 \), then prove that \( \cos^2θ + \cos^4θ = 1 \).