1. If \(x \tan 30^\circ + y \cot 60^\circ = 0\) and \(2x - y \tan 45^\circ = 1\), then calculate and write the values of \(x\) and \(y\).

2. If \[ x \cos 60^\circ = \frac{2 \tan 45^\circ}{1 + \tan^2 45^\circ} - \frac{1 - \tan^2 30^\circ}{1 + \tan^2 30^\circ} \] then find the value of \(x\).

3. 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) \(1\)

4. If \(x=\cfrac{\sqrt7+\sqrt3}{\sqrt7-\sqrt3}\) and \(xy=1\), then the value of \(\cfrac{x^2+xy+y^2}{x^2-xy+y^2}\) is —

(a) \(\cfrac{11}{12}\) (b) \(\cfrac{12}{11}\) (c) \(\cfrac{13}{12}\) (d) \(\cfrac{14}{13}\)

5. If \(x^2 = \sin^2 30^\circ + 4\cot^2 45^\circ - \sec^2 60^\circ\), then the value of \(x\) is —

(a) \(\pm 1\) (b) \(\pm \cfrac{1}{2}\) (c) \(\pm \cfrac{1}{\sqrt2}\) (d) \(\pm \cfrac{1}{\sqrt3}\)

6. If \(x = \cfrac{\sqrt{7} + \sqrt{3}}{\sqrt{7} - \sqrt{3}}\) and \(xy = 1\), then show that \[ \cfrac{x^2 + xy + y^2}{x^2 - xy + y^2} = \cfrac{12}{11} \]

7. If \(\tan(\theta + 15^\circ) = \sqrt{3}\), then what is the value of \(\sin \theta\)?

8. If \(2x = \sqrt{5} + 1\), then show that \(x^2 - x - 1 = 0\).

9. Find the value of \(x\) from the equation: \(x \sec^2 45^\circ \cdot \csc^2 45^\circ + 2(\sin 60^\circ + \sin 30^\circ) = \tan 60^\circ\)

10. If \(x^2 = \sin^2 30^\circ + 4\cot^2 45^\circ - \sec^2 60^\circ\), then what is the value of \(x\)?

11. If \(x(2 + \sqrt{3}) = y(2 - \sqrt{3}) = 1\), then the value of \(\frac{1}{x + 1} + \frac{1}{y + 1}\) will be—

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

12. If \(x = \frac{\sqrt{7} + \sqrt{3}}{\sqrt{7} - \sqrt{3}}\) and \(xy = 1\), then what is the value of \(\frac{x^2 + y^2 + xy}{x^2 + y^2 - xy}\)?

(a) \(\cfrac{6}{7}\) (b) \(\cfrac{12}{11}\) (c) \(\cfrac{13}{11}\) (d) None of the above

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

14. If \(x=\cfrac{\sqrt3+\sqrt2}{\sqrt3-\sqrt2}\) and \(xy=1\), then find the value of \(x^2 - xy + y^2\).

15. If \(x = \cfrac{\sqrt3 - \sqrt2}{\sqrt3 + \sqrt2}\) and \(xy = 1\), then what is the value of \(3x^2 - 5xy + 3y^2\)?

16. If \(x=\cfrac{\sqrt5+1}{\sqrt5-1}\) and \(xy=1\), then find the value of \(\cfrac{3x^2+5xy+3y^2}{3x^2-5xy+3y^2}\).

17. If \(x^2 = \sin^2 30^\circ + 4\cot^2 45^\circ - \sec^2 60^\circ\), then find the value of \(x\).

18. If \(x = \frac{\sqrt{7} + \sqrt{3}}{\sqrt{7} - \sqrt{3}}\) and \(xy = 1\), then show that \[ \frac{x^2 + xy + y^2}{x^2 - xy + y^2} = \frac{12}{11} \]

19. If \(x + y = \sqrt{7}\) and \(xy = 1\), then what is the value of \(x - y\)?

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

21. If \(x = \sqrt{5} + 1\), then show that \(x^2 - 2x = 4\).

22. If \(x = 3 + 2\sqrt{2}\) and \(xy = 1\), then what is the value of \(\cfrac{x^2 + y^2}{xy}\)?

23. If \(x^2 = \sin^2 30^\circ + 4\cot^2 45^\circ - \sec^2 60^\circ\), then find the value of \(x\).

24. If \(x (4 - \sqrt{3}) = y (4 + \sqrt{3}) = 1\), then the value of \(x^2 + y^2\) will be _____