Упростите выражения

1. $\sin 14^{\circ} \cos 31^{\circ} + \cos 14^{\circ} \sin 31^{\circ} = \sin(14^{\circ} + 31^{\circ}) = \sin 45^{\circ} = \frac{\sqrt{2}}{2}$ 2. $\cos(24^{\circ} + \alpha)\cos(24^{\circ} - \alpha) + \sin(24^{\circ} + \alpha)\sin(24^{\circ} - \alpha) = \cos((24^{\circ} + \alpha) - (24^{\circ} - \alpha)) = \cos(24^{\circ} + \alpha - 24^{\circ} + \alpha) = \cos(2\alpha)$ 3. $\frac{\sin 21^{\circ} \cos 28^{\circ} + \cos 21^{\circ} \sin 28^{\circ}}{\cos 18^{\circ} \cos 31^{\circ} - \sin 18^{\circ} \sin 31^{\circ}} = \frac{\sin(21^{\circ} + 28^{\circ})}{\cos(18^{\circ} + 31^{\circ})} = \frac{\sin 49^{\circ}}{\cos 49^{\circ}} = \text{tg} 49^{\circ}$ 4. $\frac{\text{tg} 2^{\circ} - \text{tg} 47^{\circ}}{1 + \text{tg} 2^{\circ} \text{tg} 47^{\circ}} = \text{tg}(2^{\circ} - 47^{\circ}) = \text{tg}(-45^{\circ}) = -\text{tg} 45^{\circ} = -1$ 5. $\frac{\text{tg}\left(\frac{\pi}{6} + \alpha\right) + \text{tg}\left(\frac{\pi}{6} - \alpha\right)}{1 - \text{tg}\left(\frac{\pi}{6} + \alpha\right) \text{tg}\left(\frac{\pi}{6} - \alpha\right)} = \text{tg}\left(\left(\frac{\pi}{6} + \alpha\right) + \left(\frac{\pi}{6} - \alpha\right)\right) = \text{tg}\left(\frac{\pi}{6} + \alpha + \frac{\pi}{6} - \alpha\right) = \text{tg}\left(\frac{2\pi}{6}\right) = \text{tg}\left(\frac{\pi}{3}\right) = \sqrt{3}$