2025 NOV TZ1 P1 Q9

1. Show that \(\frac{2\tan{\theta}}{1 + \tan^2{\theta}}\) can be expressed as \(\sin{2\theta}\)

   [3]

2. Hence, solve the equation \(\frac{2\tan{\theta}}{1 + \tan^2{\theta}} = \frac{1}{2}\) for \(0^{\circ} \leq \theta \leq 180^{\circ}\).

   [5]

Let \(x = \tan{ \left(22 \frac{1}{2} ^{\circ} \right)}\).

3. Use the result from part (a) to show that \(x^2 - 2\sqrt{2}x + 1 = 0\).

   [4]

4. Hence, find the exact value of \(\tan{ \left( 22 \frac{1}{2}^{\circ} \right) } \).

   [3]