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

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

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

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

5. Given \(x = a \sec\theta\) and \(y = b \tan\theta\), prove that \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\).

6. 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: \[ 2y = \sqrt{1 + 4x} - 1 \]