1. If \( \sin \alpha = m \sin \beta \) and \( \tan \alpha = n \tan \beta \), then prove that \[ \cos^2 \alpha = \cfrac{m^2 - 1}{n^2 - 1} \]

2. If \(x = a\sinθ\) and \(y = b\tanθ\), then prove that \(\cfrac{a^2}{x^2} - \cfrac{b^2}{y^2} = 1\)

3. If \( x = a\sinθ \) and \( y = b\tanθ \), then prove that \( \cfrac{a^2}{x^2} - \cfrac{b^2}{y^2} = 1 \).

4. 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) \(\cfrac{x^2-y^2}{x^2+y^2}\)

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

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

7. 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) \(\cfrac{y^2-x^2}{x^2+y^2}\) (c) \(\cfrac{x^2+y^2}{y^2-x^2}\) (d) None of the above

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

9. If \(\sin 17^\circ = \frac{x}{y}\), then prove that \[ \sec 17^\circ - \sin 73^\circ = \frac{x^2}{y \sqrt{y^2 - x^2}} \]

10. If \(\sin 17^\circ = \frac{x}{y}\), then prove that \[ \sec 17^\circ - \sin 73^\circ = \frac{x^2}{y\sqrt{y^2 - x^2}} \]

11. ```html \(\cfrac{\sinθ}{x}=\cfrac{\cosθ}{y}\)
i.e., \(\cfrac{\sinθ}{\cosθ}=\cfrac{x}{y}\)
i.e., \(\tanθ=\cfrac{x}{y}\)
i.e., \(\tan^2θ=\cfrac{x^2}{y^2}\)
i.e., \(1+ \tan^2θ=1+\cfrac{x^2}{y^2}\)
i.e., \(\sec^2θ=\cfrac{y^2+x^2}{y^2}\)
i.e., \(\secθ=\cfrac{\sqrt{y^2+x^2}}{y}\)
i.e., \(\cosθ=\cfrac{y}{\sqrt{x^2+y^2}} \)

Now, in the equation \(\cfrac{\sinθ}{x}=\cfrac{\cosθ}{y}\),
substituting \(\cosθ=\cfrac{y}{\sqrt{x^2+y^2}}\), we get
\(\cfrac{\sinθ}{x}=\cfrac{\cfrac{y}{\sqrt{x^2+y^2}}}{y}\)
i.e., \(\cfrac{\sinθ}{x}=\cfrac{1}{\sqrt{x^2+y^2}}\)
i.e., \(\sinθ=\cfrac{x}{\sqrt{x^2+y^2}}\)

∴ \(\sinθ−\cosθ=\cfrac{x}{\sqrt{x^2+y^2}}−\cfrac{y}{\sqrt{x^2+y^2}}\)
\(=\cfrac{x−y}{\sqrt{x^2+y^2}}\) (Proved) ```

12. If \( \sin 17^\circ = \frac{x}{y} \), then prove that: \[ \sec 17^\circ - \sin 73^\circ = \frac{x^2}{y\sqrt{y^2 - x^2}} \]

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

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

15. Given \( \sin\alpha = m\sin\beta \) and \( \tan\alpha = n\tan\beta \), prove that \[ \sin\alpha = \sqrt{\frac{n^2 - m^2}{n^2 - 1}} \]

16. If \( \sin 18^\circ = \frac{x}{y} \), then show that \[ \sec 18^\circ - \sin 72^\circ = \frac{x^2}{y\sqrt{y^2 - x^2}} \]

17. If \(x = cy + bz\), \(y = az + cx\), and \(z = bx + ay\), then prove that \[ \frac{x^2}{1 - a^2} = \frac{y^2}{1 - b^2} \]

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 \(x = cy + bz\), \(y = az + cx\), and \(z = bx + ay\), then prove that \[ \frac{x^2}{1 - a^2} = \frac{y^2}{1 - b^2} = \frac{z^2}{1 - c^2} \]

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

21. If \(\cfrac{x − x\tan^2 30°}{1 + \tan^2 30°} = \sin^2 30° + 4\cos^2 45° - \sec^2 60°\), find the value of \(x\). Let me know if you'd like it solved as well.

22. 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 \(\cfrac{1}{a + b + c}\).

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

24. From \(x = a \sin \theta\) and \(y = b \tan \theta\), eliminate \(\theta\).

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

26. If \( \sin 17^\circ = \frac{x}{y} \), then show that \[ \sec 17^\circ - \sin 73^\circ = \frac{x^2}{y \sqrt{y^2 - x^2}} \]

27. A person bought \(y\) pencils for ₹\(x\). If each pencil had cost ₹1 less, then with the same amount ₹\(x\), he would have received one additional pencil. Prove that: \[ 2y = \sqrt{1 + 4x} - 1 \]

28. If \(x\) is a real positive number and \(\sin x = \frac{2}{3}\), then what is the value of \(\tan x\)?

(a) \(\cfrac{2}{\sqrt5}\) (b) \(\cfrac{\sqrt5}{2}\) (c) \(\sqrt{\cfrac{5}{3}}\) (d) \(\cfrac{\sqrt5}{\sqrt2}\)

29. If \( \cosθ = \frac{x}{\sqrt{x^2 + y^2}} \), then prove that \( x\sinθ = y\cosθ \).

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