1. If \[ \frac{a}{y + z} = \frac{b}{z + x} = \frac{c}{x + y} \] then prove that \[ \frac{a(b - c)}{y^2 - z^2} = \frac{b(c - a)}{z^2 - x^2} = \frac{c(a - b)}{x^2 - y^2} \]

2. Given: \[ x^2 : (by + cz) = y^2 : (cz + ax) = z^2 : (ax + by) = 1 \] Show that: \[ \frac{a}{a + x} + \frac{b}{b + y} + \frac{c}{c + z} = 1 \]

3. If \[ \frac{b + c - a}{y + z - x} = \frac{c + a - b}{z + x - y} = \frac{a + b - c}{x + y - z} \] then prove that \[ \frac{a}{x} = \frac{b}{y} = \frac{c}{z} \]

4. If \[ \frac{by + cz}{b^2 + c^2} = \frac{cz + ax}{c^2 + a^2} = \frac{ax + by}{a^2 + b^2} \] then prove that \[ \frac{x}{a} = \frac{y}{b} = \frac{z}{c} \]

5. If \[ \frac{x^2 - yz}{a} = \frac{y^2 - zx}{b} = \frac{z^2 - xy}{c} \] then prove that \[ (a + b + c)(x + y + z) = ax + by + cz \]

6. If \[ \frac{x^2}{by + cz} = \frac{y^2}{cz + ax} = \frac{z^2}{ax + by} = 1 \] then prove that \[ \frac{a}{a + x} + \frac{b}{b + y} + \frac{c}{c + z} = 1 \]

7. If \[ \cfrac{x}{b + c - a} = \cfrac{y}{c + a - b} = \cfrac{z}{a + b - c} \] then prove that \[ (b - c)x + (c - a)y + (a - b)z = 0 \]

8. If \[ \cfrac{x}{b + c} = \cfrac{y}{c + a} = \cfrac{z}{a + b} \] then prove that \[ \cfrac{a}{y + z - x} = \cfrac{b}{z + x - y} = \cfrac{c}{x + y - z} \]

9. If \[ \cfrac{x + y}{3a - b} = \cfrac{y + z}{3b - c} = \cfrac{z + x}{3c - a} \] then prove that \[ \cfrac{x + y + z}{a + b + c} = \cfrac{ax + by + cz}{a^2 + b^2 + c^2} \]

10. If \[ \cfrac{x^2}{by + cz} = \cfrac{y^2}{cz + ax} = \cfrac{z^2}{ax + by} = 1 \] then show that \[ \cfrac{a}{a + x} + \cfrac{b}{b + y} + \cfrac{c}{c + z} = 1 \]

11. If \[ \cfrac{x^2 - yz}{a} = \cfrac{y^2 - zx}{b} = \cfrac{z^2 - xy}{c}, \] then prove that \[ (a + b + c)(x + y + z) = ax + by + cz. \]

12. If \[ \cfrac{x^2 - yz}{a} = \cfrac{y^2 - zx}{b} = \cfrac{z^2 - xy}{c}, \] then prove that \[ (a + b + c)(x + y + z) = ax + by + cz. \]

13. If \[ \cfrac{a}{y + z} = \cfrac{b}{z + x} = \cfrac{c}{x + y}, \] then prove that \[ \cfrac{a(b - c)}{y^2 - z^2} = \cfrac{b(c - a)}{z^2 - x^2} = \cfrac{c(a - b)}{x^2 - y^2}. \]

14. If \[ x^2 : (by + cz) = y^2 : (cz + ax) = z^2 : (ax + by) = 1 \] then show that \[ \frac{a}{a + x} + \frac{b}{b + y} + \frac{c}{c + z} = 1 \]

15. If \[ \cfrac{x}{xa + yb + zc} = \cfrac{y}{ya + zb + xc} = \cfrac{z}{za + xb + yc} \] and \(x + y + z \ne 0\), then show that each of the above ratios is equal to \(\cfrac{1}{a + b + c}\).

16. If \((b + c - a)x = (c + a - b)y = (a + b - c)z = 2\), then show that \[ \left(\frac{1}{x} + \frac{1}{y}\right)\left(\frac{1}{y} + \frac{1}{z}\right)\left(\frac{1}{z} + \frac{1}{x}\right) = abc \]

17. If \(bcx = cay = abz\), then prove that \[ \frac{ax + by}{a^2 + b^2} = \frac{by + cz}{b^2 + c^2} = \frac{cz + ax}{c^2 + a^2} \]

18. If \(a^2 = by + cz\), \(b^2 = cz + ax\), and \(c^2 = ax + by\), then prove that \[ \frac{x}{a + x} + \frac{y}{b + y} + \frac{z}{c + z} = 1 \]

19. If \[ \frac{2x}{3} = \frac{4y}{5} = \frac{7z}{9} \] then find the value of \[ \frac{4x + 12y - 21z}{3y} \]

20. If \[ \frac{3 - 5x}{x} + \frac{3 - 5y}{y} + \frac{3 - 5z}{z} = 0 \] then what is the value of \[ \frac{1}{x} + \frac{1}{y} + \frac{1}{z}? \]

21. If \(x = 2\), \(y = 3\), and \(z = 6\), then calculate the value of \[ \frac{3\sqrt{x}}{\sqrt{y} + \sqrt{z}} - \frac{4\sqrt{y}}{\sqrt{z} + \sqrt{x}} + \frac{\sqrt{z}}{\sqrt{x} + \sqrt{y}} \]

22. If \[ (a+b+c)x = (b+c-a)y = (c+a-b)z = (a+b-c)w \] prove that \[ \cfrac{1}{y}+\cfrac{1}{z}+\cfrac{1}{w}=\cfrac{1}{x} \]

23. If \(\cfrac{a}{b} = \cfrac{x}{y}\), then prove that \[ (a + b)(a^2 + b^2)x^3 = (x + y)(x^2 + y^2)a^3 \]

24. If \[ \cfrac{x}{lm - n^2} = \cfrac{y}{mn - l^2} = \cfrac{z}{nl - m^2} \] then prove that \[ lm - my + nz = 0 \]

25. 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 \(-1\).

26. If \(x = 2\), \(y = 3\), and \(z = 6\), then calculate the value of \[ \frac{3\sqrt{x}}{\sqrt{y} + \sqrt{z}} - \frac{4\sqrt{y}}{\sqrt{z} + \sqrt{x}} + \frac{\sqrt{z}}{\sqrt{x} + \sqrt{y}} \]

27. If \[ (b + c - a) x = (c + a - b) y = (a + b - c) z = 2 \] then prove that \[ \left( \frac{1}{y} + \frac{1}{z} \right) \left( \frac{1}{z} + \frac{1}{x} \right) \left( \frac{1}{x} + \frac{1}{y} \right) = abc \]

28. If \( \tan \theta = \frac{x}{y} \), then what is the value of \[ \frac{x\sin\theta - y\cos\theta}{x\sin\theta + y\cos\theta}? \]

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

29. If \(\frac{x}{y} = \frac{a + 2}{a - 2}\), then prove that \[ \frac{x^2 - y^2}{x^2 + y^2} = \frac{4a}{a^2 + 4} \]

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