Тригонометрические выражения

$$\sin^4\alpha+\sin^2\alpha\cos^2\alpha-\sin^2\alpha=\sin^2\alpha(\sin^2\alpha+\cos^2\alpha-1)=0$$

$$\text{tg}^2\beta+\text{ctg}^2\beta+2-\frac{1}{\cos^2\beta}=\text{tg}^2\beta+\text{ctg}^2\beta+2-(1+\text{tg}^2\beta\text{ctg}^2\beta+1=\frac{1}{\sin^2\beta}$$

$$\text{ctg}(-\alpha)\cdot\text{tg}(-\alpha)-\sin^2(-\alpha)=1-\sin^2(-\alpha)=\cos^2(-\alpha)=\cos^2\alpha$$

$$(\text{tg}\alpha+\text{ctg}\alpha)^2=\text{tg}^2\alpha+\text{ctg}^2\alpha+2\text{tg}\alpha\text{ctg}\alpha$$
$$4=\text{tg}^2\alpha+\text{ctg}^2\alpha+2\cdot1$$
$$\text{tg}^2\alpha+\text{ctg}^2\alpha=2$$

$$\frac{\sin\alpha-\cos\alpha}{\sin\alpha+\cos\alpha}=\frac{-\sqrt{2}\sin(45^{\circ}-\alpha)}{\sqrt{2}\cos(45^{\circ}-\alpha)}=\text{tg}(\alpha-45^{\circ})=\frac{\text{tg}\alpha-\text{tg}45^{\circ}}{1+\text{tg}\alpha\text{tg}45^{\circ}}=\frac{3-1}{1+3\cdot1}=\frac{1}{2}$$

$$3\sin^2\alpha-7\cos^2\alpha=-[3(\cos^2\alpha-\sin^2\alpha)+4\cos^2\alpha]=-[3(2\cos^2\alpha-1+4\cos^2\alpha)]=3-10\cos^2\alpha=3-\frac{10}{100}=2.9$$

$$\cos(\alpha-\beta+\frac{\pi}{2})+2\sin(\alpha+\pi)\cos(\beta-\pi)=-\sin(\alpha-\beta)+2(-\sin\alpha)(-\cos\beta)=$$
$$=-\sin\alpha\cos\beta+\cos\alpha\sin\beta+2\sin\alpha\cos\beta=\sin\alpha\cos\beta+\cos\alpha\sin\beta=$$
$$=\sin(\alpha+\beta)=\sin0.25\pi=\sin\frac{\pi}{4}=\frac{\sqrt{2}}{2}$$

$$\text{tg}7^{\circ}\text{tg}83^{\circ}+\text{tg}19^{\circ}\text{tg}71^{\circ}=\text{tg}7^{\circ}\text{tg}(90^{\circ}-7^{\circ})+\text{tg}19^{\circ}\text{tg}(90^{\circ}-19^{\circ})=\text{tg}7^{\circ}\text{ctg}7^{\circ}+\text{tg}19^{\circ}\text{ctg}19^{\circ}=1+1=2$$

$$\cos^4\frac{\alpha}{2}-\sin^4\frac{\alpha}{2}=(\cos^2\frac{\alpha}{2}-\sin^2\frac{\alpha}{2})(\cos^2\frac{\alpha}{2}+\sin^2\frac{\alpha}{2})=\cos2\frac{\alpha}{2}\cdot1=\cos\alpha$$

$$\frac{\sin40^{\circ}}{2\cos^2 20^{circ}}=\frac{2\sin20^{\circ}\cos20^{\circ}}{2\cos^2 20^{circ}}=\text{tg}20^{\circ}$$

$$\sin48^{\circ}+\sin12^{\circ}=2\sin30^{\circ}\cos18^{\circ}=\cos18^{\circ}$$

$$\sin(\text{arccos}\frac{1}{4})=$$…
$$\text{arccos}\frac{1}{4}=t\in(0;\frac{\pi}{2})$$ – I четверть
$$\cos t=\frac{1}{4}$$
$$\sin t=+\sqrt{1-\frac{1}{16}}=\frac{\sqrt{15}}{4}$$

$$\text{arcsin}\frac{1}{6}=t\in(0;\frac{\pi}{2})$$
$$\sin t=\frac{1}{6}$$
$$\cos2t=1-2\sin^2t=1-2\cdot\frac{1}{36}=\frac{17}{18}$$

$$\text{arcsin}0.6=t\in(0;\frac{\pi}{2})$$ – I четверть
$$\sin t= 0.6$$
$$\text{tg}(-\frac{1}{2}t)=-\text{tg}\frac{t}{2}=-\frac{1-\cos t}{\sin t}$$
$$\cos t=\sqrt{1-0.36}=0.8$$
$$\text{tg}(-\frac{1}{2}\text{arcsin}0.6)=-\frac{1-0.8}{0.6}=-\frac{1}{3}$$

1. $$\sin510^{\circ}=\sin(360^{\circ}+90^{\circ}+60^{\circ})=\sin(90^{\circ}+60^{\circ})=\cos60^{\circ}=\frac{1}{2}$$

2. $$\cos690^{\circ}=\cos(360^{\circ}+270^{\circ}+60^{\circ})=\sin60^{\circ}=\frac{\sqrt{3}}{2}$$

3. $$\cos840^{\circ}=\cos(2\cdot360^{\circ}+90^{\circ}+30^{\circ})=-\sin30^{\circ}=-\frac{1}{2}$$

4. $$\sin960^{\circ}=\sin(2\cdot360^{\circ}+180^{\circ}+60^{\circ})=-\sin60^{\circ}=-\frac{\sqrt{3}}{2}$$