1. If \(\sin^4 \theta + \sin^2 \theta = 1\), 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 \(\cfrac{a^2}{b+c} = \cfrac{b^2}{c+a} = \cfrac{c^2}{a+b} = 1\), then prove that \(\cfrac{1}{a+1} + \cfrac{1}{b+1} + \cfrac{1}{c+1} = 1\).

4. If \(a + \frac{1}{b} = 1\) and \(b + \frac{1}{c} = 1\), then prove that \(c + \frac{1}{a} = 1\).

5. If \(\frac{a^2}{b + c} = \frac{b^2}{c + a} = \frac{c^2}{a + b} = 1\), then prove that \[\frac{1}{1 + a} + \frac{1}{1 + b} + \frac{1}{1 + c} = 1\]

6. If \[ \frac{a}{b + c} = \frac{b}{c + a} = \frac{c}{a + b} \] then prove that each of these ratios is either \(\frac{1}{2}\) or \(-1\).

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

8. If \(a + \frac{1}{b} = 1\) and \(b + \frac{1}{c} = 1\), then show that \(c + \frac{1}{a} = 1\).

9. If \(\cfrac{a^2}{b+c} = \cfrac{b^2}{c+a} + \cfrac{c^2}{a+b} = 1\), then prove that \[ \cfrac{1}{1+a} + \cfrac{1}{1+b} + \cfrac{1}{1+c} = 1 \]

10. If \(a = \cfrac{\sqrt7+\sqrt3}{\sqrt7-\sqrt3}\) and \(ab = 1\), then prove that \(\cfrac{a^2+ab+b^2}{a^2-ab+b^2}=\cfrac{12}{11}\).

11. If \(a + c = 2b\) and \(q^2 = pr\), then prove that \(p^{b-c} \cdot q^{c-a} \cdot r^{a-b} = 1\).

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

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

14. If \(\cfrac{a}{1-a}+\cfrac{b}{1-b}+\cfrac{c}{1-c}=1\), then prove that \(\cfrac{1}{1-a}+\cfrac{1}{1-b}+\cfrac{c}{1-c}=4\).

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

16. If \(\cfrac{a^2}{b+c} = \cfrac{b^2}{c+a} + \cfrac{c^2}{a+b} = 1\), prove that \(\cfrac{1}{1+a} + \cfrac{1}{1+b} + \cfrac{1}{1+c} = 1\).

17. If \(\cfrac{a^2}{b+c} = \cfrac{b^2}{c+a} + \cfrac{c^2}{a+b} = 1\), then show that \(\cfrac{1}{1+a} + \cfrac{1}{1+b} + \cfrac{1}{1+c} = 1\).

18. If \(\cfrac{x}{y+z}=\cfrac{y}{z+x}=\cfrac{z}{x+y}\), then prove that the value of each ratio is equal to either \(\cfrac{1}{2}\) or \(-1\).

19. If \(a : b = b : c\), then prove that \(\cfrac{abc(a+b+c)^3}{(ab+bc+ca)^3} = 1\).

20. If \[ \frac{x}{y + z} = \frac{y}{z + x} = \frac{z}{x + y} \] then prove that the value of each ratio is either \(\frac{1}{2}\) or \(-1\).

21. If \(x^2 : (by + cz) = y^2 : (cz + ax) = z^2 : (ax + by) = 1\), then show that \(\cfrac{a}{a + x} + \cfrac{b}{b + y} + \cfrac{c}{c + z} = 1\).