502 1) 2 sin 75° * cos 75°; 2) cos² 75° - sin² 75°; 3) (6 tg 75°) / (1 - tg² 75°); 4) (tg² 22°30' - 1) / (tg 22°30')

502. Используем формулы двойного угла: $\sin 2\alpha = 2 \sin \alpha \cos \alpha$, $\cos 2\alpha = \cos^2 \alpha - \sin^2 \alpha$, $\text{tg} 2\alpha = \frac{2 \text{tg} \alpha}{1 - \text{tg}^2 \alpha}$. 1) $2 \sin 75^\circ \cos 75^\circ = \sin (2 \cdot 75^\circ) = \sin 150^\circ = \sin(180^\circ - 30^\circ) = \sin 30^\circ = \frac{1}{2}$. 2) $\cos^2 75^\circ - \sin^2 75^\circ = \cos (2 \cdot 75^\circ) = \cos 150^\circ = \cos(180^\circ - 30^\circ) = -\cos 30^\circ = -\frac{\sqrt{3}}{2}$. 3) $\frac{6 \text{tg} 75^\circ}{1 - \text{tg}^2 75^\circ} = 3 \cdot \frac{2 \text{tg} 75^\circ}{1 - \text{tg}^2 75^\circ} = 3 \text{tg} 150^\circ = 3 \text{tg}(180^\circ - 30^\circ) = -3 \text{tg} 30^\circ = -3 \cdot \frac{\sqrt{3}}{3} = -\sqrt{3}$. 4) $\frac{\text{tg}^2 22^\circ 30' - 1}{\text{tg} 22^\circ 30'} = -\frac{1 - \text{tg}^2 22^\circ 30'}{\text{tg} 22^\circ 30'} = -\frac{2}{\text{tg} (2 \cdot 22^\circ 30')} = -\frac{2}{\text{tg} 45^\circ} = -\frac{2}{1} = -2$. 503. Вычислить $\sin 2\alpha = 2 \sin \alpha \cos \alpha$. 1) $\sin \alpha = \frac{3}{5}$, $\frac{\pi}{2} < \alpha < \pi$ (II четверть, $\cos \alpha < 0$). $\cos \alpha = -\sqrt{1 - \sin^2 \alpha} = -\sqrt{1 - (3/5)^2} = -4/5$. $\sin 2\alpha = 2 \cdot \frac{3}{5} \cdot \left(-\frac{4}{5}\right) = -\frac{24}{25} = -0,96$. 2) $\cos \alpha = -\frac{4}{5}$, $\pi < \alpha < \frac{3\pi}{2}$ (III четверть, $\sin \alpha < 0$). $\sin \alpha = -\sqrt{1 - \cos^2 \alpha} = -\sqrt{1 - (-4/5)^2} = -3/5$. $\sin 2\alpha = 2 \cdot \left(-\frac{3}{5}\right) \cdot \left(-\frac{4}{5}\right) = \frac{24}{25} = 0,96$. 504. Вычислить $\cos 2\alpha$. 1) $\cos \alpha = \frac{4}{5} \Rightarrow \cos 2\alpha = 2\cos^2 \alpha - 1 = 2 \cdot \left(\frac{4}{5}\right)^2 - 1 = \frac{32}{25} - 1 = \frac{7}{25} = 0,28$. 2) $\sin \alpha = -\frac{3}{5} \Rightarrow \cos 2\alpha = 1 - 2\sin^2 \alpha = 1 - 2 \cdot \left(-\frac{3}{5}\right)^2 = 1 - \frac{18}{25} = \frac{7}{25} = 0,28$. 505. $\text{tg} \alpha = 0,5$. Вычислить $\text{tg} 2\alpha = \frac{2 \text{tg} \alpha}{1 - \text{tg}^2 \alpha} = \frac{2 \cdot 0,5}{1 - 0,5^2} = \frac{1}{1 - 0,25} = \frac{1}{0,75} = \frac{4}{3} = 1\frac{1}{3}$. Ответ: 502: 1) 0,5; 2) $-\frac{\sqrt{3}}{2}$; 3) $-\sqrt{3}$; 4) -2. 503: 1) -0,96; 2) 0,96. 504: 1) 0,28; 2) 0,28. 505: $1\frac{1}{3}$.