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

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

3. If \(\sin\theta+\sin^2\theta=1\), prove that \(\cos^2\theta+\cos^4\theta=1\).

4. If \(a(\tan\theta + \cot\theta) = 1\) and \(\sin\theta + \cos\theta = b\), then prove that \(2a = b^2 - 1\), where \(0^\circ < \theta < 90^\circ\).

5. If \(sin\theta + sin^2\theta = 1\), prove that \(cos^2\theta + cos^4\theta = 1\).

6. If \(sin\theta + sin^2\theta = 1\), prove that \(cos^2\theta + cos^4\theta = 1\).

7. If \( \cot\theta = \frac{x}{y} \), then prove that \[ \frac{x\cos\theta - y\sin\theta}{x\cos\theta + y\sin\theta} = \frac{x^2 - y^2}{x^2 + y^2} \]

8. If \(\cos\theta = \frac{x}{\sqrt{x^2 + y^2}}\), then prove that \(x \sin \theta = y \cos \theta\).

9. If \(x = a\cos(90^\circ - \theta)\), \(y = b\cot(90^\circ - \theta)\), then prove that \(\cfrac{a^2}{x^2} - \cfrac{b^2}{y^2} = 1\).

10. If \(a \sin^2 \theta + b \cos^2 \theta = c\), then what is the value of \(\tan^2 \theta\)?

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

11. If \(\cfrac{\sin \theta}{x} = \cfrac{\cos \theta}{y}\), then prove that \( \sin \theta - \cos \theta = \cfrac{x - y}{\sqrt{x^2 + y^2}} \).

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

13. If \( \cos\theta + \sqrt{3} \sin\theta = 2 \sin\theta \), then prove that \( \sin\theta - \sqrt{3} \cos\theta = 2 \cos\theta \).

14. If \(x = a \cos\theta + b \sin\theta\) and \(y = a \sin\theta - b \cos\theta\), then prove that \(x^2 + y^2 = a^2 + b^2\).

15. If \(\cfrac{\sin\theta}{x} = \cfrac{\cos\theta}{y}\), then prove that \(\sin\theta - \cos\theta = \cfrac{x - y}{\sqrt{x^2 + y^2}}\).

16. In a right-angled triangle, if the ratio of the perpendicular (opposite side) to the hypotenuse with respect to a positive acute angle \(\theta\) is \(12 : 13\), then determine the ratio of the perpendicular to the base and the ratio of the hypotenuse to the base, and verify that \( \sec^2\theta = 1 + \tan^2\theta \).

17. If \( \sin\theta + \csc\theta = 2 \), then prove that \[ \sin^{99}\theta + \csc^{99}\theta = 2 \]

18. If \( \sin\theta + \sin^2\theta = 1 \), prove that \( \cos^2\theta + \cos^4\theta = 1 \).

19. If \( \sin\theta + \sin^2\theta = 1 \), then prove: \[ \cos^2\theta + \cos^4\theta = 1 \]

20. 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

21. If \( \sin\theta + \csc\theta = 2 \), then what is the value of \( \sin^2\theta + \csc^2\theta \)?

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

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

23. Prove that \[ \sqrt{\cfrac{1 + \cos \theta}{1 - \cos \theta}} = \csc \theta + \cot \theta \]

24. 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 \(0^\circ < \theta < 90^\circ\).

25. If \(\tan 2\theta \cdot \tan 3\theta = 1\), then find the value of \(\theta\), given that \(0 \leq \theta \leq \cfrac{\pi}{2}\).

26. Given: \(\tan \theta + \sin \theta = m\) and \(\tan \theta - \sin \theta = n\) Prove that: \[ m^2 - n^2 = 4\sqrt{mn} \quad \text{where } 0^\circ < \theta < 90^\circ \]

27. If \(x = \frac{1 + \sin\theta}{\cos\theta}\), then show that \(x^2 - 2x\tan\theta - 1 = 0\).

28. 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) \(1\)

29. Determine the values of \(\theta\) for which \(\sin^2\theta - 3\sin\theta + 2 = 0\) holds true, given that \(0^\circ < \theta < 90^\circ\).

30. Prove that \[ \frac{\tan \theta + \sec \theta - 1}{\tan \theta - \sec \theta + 1} = \frac{1 + \sin \theta}{\cos \theta} \]