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

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

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

4. Find the minimum value of the expression \((4\csc^2 θ + 9\sin^2 θ)\) for \(0° < θ < 90°\).

5. If \( a \cosθ + b \sinθ = 4 \) and
\( a \sinθ - b \cosθ = 3 \), then \( a^2 + b^2 \) = ?

(a) 7 (b) 12 (c) 25 (d) 32

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

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

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

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

10. If \(0^\circ < \theta < 90^\circ\), then \(\sin\theta < \sin^2\theta\).

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

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

13. If \( \cos^2θ - \sin^2θ = \tan^2α \), then prove that \( \cos^2α - \sin^2α = \tan^2θ \).

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

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

16. If \(\sin \theta = \cfrac{p^2 - q^2}{p^2 + q^2}\), then show that \(\cot \theta = \cfrac{2pq}{p^2 - q^2}\) where \(p > q\) and \(0^\circ < \theta < 90^\circ\).

17. If \(\sin^4θ + \sin^2θ = 1\), then prove that \(\tan^4θ - \tan^2θ = 1\).

18. 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) \(\cfrac{1}{7}\)

19. If \( \tan 4θ \cdot \tan 6θ = 1 \), then determine the value of \( θ \) given that \( 0° < θ < 90° \).

(a) 5° (b) 4° (c) 9° (d) 3°

20. If \(0^\circ < \theta < 90^\circ\), then \( \cos \theta > \cos^2 \theta \).

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

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

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

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

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

26. If \(0^\circ < \theta \leq 90^\circ\), then what is the minimum value of \((4\csc^2\theta + 9\sin^2\theta)\)?

27. ```html \(\cfrac{\sinθ}{x}=\cfrac{\cosθ}{y}\)
i.e., \(\cfrac{\sinθ}{\cosθ}=\cfrac{x}{y}\)
i.e., \(\tanθ=\cfrac{x}{y}\)
i.e., \(\tan^2θ=\cfrac{x^2}{y^2}\)
i.e., \(1+ \tan^2θ=1+\cfrac{x^2}{y^2}\)
i.e., \(\sec^2θ=\cfrac{y^2+x^2}{y^2}\)
i.e., \(\secθ=\cfrac{\sqrt{y^2+x^2}}{y}\)
i.e., \(\cosθ=\cfrac{y}{\sqrt{x^2+y^2}} \)

Now, in the equation \(\cfrac{\sinθ}{x}=\cfrac{\cosθ}{y}\),
substituting \(\cosθ=\cfrac{y}{\sqrt{x^2+y^2}}\), we get
\(\cfrac{\sinθ}{x}=\cfrac{\cfrac{y}{\sqrt{x^2+y^2}}}{y}\)
i.e., \(\cfrac{\sinθ}{x}=\cfrac{1}{\sqrt{x^2+y^2}}\)
i.e., \(\sinθ=\cfrac{x}{\sqrt{x^2+y^2}}\)

∴ \(\sinθ−\cosθ=\cfrac{x}{\sqrt{x^2+y^2}}−\cfrac{y}{\sqrt{x^2+y^2}}\)
\(=\cfrac{x−y}{\sqrt{x^2+y^2}}\) (Proved) ```

28. If \(0^\circ < \theta < 90^\circ\), then show that \(\sin\theta + \cos\theta > 1\)

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

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

30. If \( \cosθ = \frac{x}{\sqrt{x^2 + y^2}} \), then prove that \( x\sinθ = y\cosθ \).