1. If \(\frac{x}{y} \propto (x + y)\) and \(\frac{y}{x} \propto (x - y)\), then find the value of \(x^2 - y^2\).

(a) "Proportional to \(x\)" (b) "Proportional to \(y\)" (c) "Proportional to \(xy\)" (d) constant

2. Given \(x = 3 + \sqrt{3}\) and \(y = 6\), find the value of \((x + y)^2\).

3. From the equation \(5 \sin^2 \theta + 4 \cos^2 \theta = \frac{9}{2}\), find the value of \(\tan \theta\), where \(0^\circ < \theta < 90^\circ\).

4. If \(\tan \alpha = \cot \beta\), find the value of \(\cos(\alpha + \beta)\), where \(0^\circ < \alpha, \beta < 90^\circ\).

5. Given: \(\tan A = \frac{x}{y}\), find the value of \(\frac{\cos A - \sin A}{\cos A + \sin A}\).

6. If \(x = \sqrt{3} + \sqrt{2}\) and \(y = \cfrac{1}{\sqrt{3} + \sqrt{2}}\), then find the value of \((x + y)^2 + (x - y)^2\).

7. If \(x:y = 3:4\), find the value of \((3y - x) : (2x + y)\).

8. If \((3x - 2y) : (x + 3y) = 5 : 6\), then find the value of \((2x - 5y) : (3x + 4y)\).

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

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

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

12. 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 \(\tan θ\).

13. If \(\cot θ = 2\), then find the values of \(\tan θ\) and \(\sec θ\), and show that: \[1 + \tan^2θ = \sec^2θ\]

14. If \(\cfrac{\sin \theta + \cos \theta}{\sin \theta - \cos \theta} = 7\), then find the value of \(\tan \theta\).

15. Given: \[ x = 3 + \sqrt{3}, \quad xy = 6 \] Find the value of \((x + y)\).