1. If \( \cosθ = \cfrac{x}{\sqrt{x^2 + y^2}} \), then show that \( x\sinθ = y\cosθ \).

2. If \(\cos\theta = \frac{x}{\sqrt{x^2 + y^2}}\), then prove that \(x \sin \theta = y \cos \theta\).

3. If \(\cfrac{\sin \theta}{x} = \cfrac{\cos \theta}{y}\), then prove that \( \sin \theta - \cos \theta = \cfrac{x - y}{\sqrt{x^2 + y^2}} \).

4. If \(\cfrac{\sin\theta}{x} = \cfrac{\cos\theta}{y}\), then prove that \(\sin\theta - \cos\theta = \cfrac{x - y}{\sqrt{x^2 + y^2}}\).

5. If \(\cfrac{a + b − c}{a + b} = \cfrac{b + c − a}{b + c} = \cfrac{c + a − b}{c + a}\) and \(a + b + c ≠ 0\), then prove that \(a = b = c\).

6. If \(\angle A + \angle B = 90^\circ\), then prove that \(1 + \frac{\tan A}{\tan B} = \csc^2 B\).

7. If \(a : b = c : d\), then prove that \((a^2 + c^2)(b^2 + d^2) = (ab + cd)^2\).

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

9. If \(\cfrac{a}{b + c} = \cfrac{b}{c + a} = \cfrac{c}{a + b}\) and \(a + b + c \ne 0\), then prove that \(a = b = c\).

10. If \(\sin^4 \theta + \sin^2 \theta = 1\), then prove that \(\tan^4 \theta - \tan^2 \theta = 1\).

11. If \(a : b = b : c\), then prove that \((a + b + c)(a - b + c) = a^2 + b^2 + c^2\).

12. If \(\sin^4θ + \sin^2θ = 1\), then prove that \(\tan^4θ - \tan^2θ = 1\).

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

14. If \(\cfrac{bz+cy}{a} = \cfrac{cx+az}{b} = \cfrac{ay+bx}{c}\), then prove that \(\cfrac{x}{a(b^2+c^2-a^2)} = \cfrac{y}{b(c^2+a^2-b^2)} = \cfrac{z}{c(a^2+b^2-c^2)}\).

15. If \(a:b=c:d\), then prove that \((a^2+c^2)(b^2+d^2)=(ab+cd)^2\).

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

17. If \(\sin^2\theta + \sin^4\theta = 1\), then prove that \(\tan^4\theta - \tan^2\theta = 1\).

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

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

20. If the roots of the quadratic equation \(ax^2+bx+c=0\) are in the ratio \(1:2\), then prove that \(2b^2=9ac\).

21. 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 \(-1\).

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

23. If \( \cfrac{\sin θ}{x} = \cfrac{\cos θ}{y} \), then show that \( \sin θ - \cos θ = \cfrac{x - y}{\sqrt{x^2 + y^2}} \).

24. If \( \sinθ + \sin^2θ = 1 \), then prove that \( \cos^2θ + \cos^4θ = 1 \).

25. If \( \cos\theta + \sqrt{3} \sin\theta = 2 \sin\theta \), then prove that \( \sin\theta - \sqrt{3} \cos\theta = 2 \cos\theta \).

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

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

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

29. 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: \[ 2y = \sqrt{1 + 4x} - 1 \]

30. If \( \cos 41^\circ = \frac{x}{\sqrt{x^2 + y^2}} \), then find the value of \( \tan 49^\circ \).