- Библиотека Кэмп

Разбор задачи

Условие:

\begin{array}{l}\sin 20^{\circ} \cdot \cos 70^{\circ}+\sin ^{2} 110^{\circ} \cdot \cos ^{2} 250^{\circ}+\sin ^{2} 290^{\circ} \cdot \cos ^{2} 340^{\circ} \\ \left(\frac{\operatorname{tg}^{2} 590^{\circ}}{\cos ^{2} 320^{\circ}}+\frac{\sin 111^{\circ}}{\cos 159^{\circ}}\right) \cdot\left(\frac{\cos 279^{\circ}}{\sin 549^{\circ}}+\frac{\operatorname{ctg}^{2} 950^{\circ}}{\sin ^{2} 400^{\circ}}\right) \\ \sin (2 \pi+t) \cdot \operatorname{ctg}(3 \pi+t)-\cos (2 \pi-t) \cdot \operatorname{tg}(3 \pi-t)=\sin t+\cos t \\ \sin 395^{\circ} \cdot \sin 505^{\circ}+\cos 575^{\circ} \cdot \cos 865^{\circ}+\operatorname{tg} 606^{\circ} \cdot \operatorname{tg} 1104^{\circ}=2 \\ \sin 405^{\circ} \cdot \cos 675^{\circ}+\operatorname{tg} 562^{\circ} \cdot \operatorname{tg} 788^{\circ}+\frac{1}{\cos 660^{\circ}} \cdot \frac{1}{\cos 1200^{\circ}}=-2,5\end{array}

Первое выражение: (\sin 20^{\circ} \cdot \cos 70^{\circ}+\sin ^{2} 110^{\circ} \cdot \cos ^{2} 250^{\circ}+\sin ^{2} 290^{\circ} \cdot \cos ^{2} 340^{\circ})
- Используем тригонометрические идентичности:
  - (\cos 70^{\circ} = \sin 20^{\circ})
  - (\sin 110^{\circ} = \cos 20^{\circ})
  - (\cos 250^{\circ} = -\sin 70^{\circ})
  - (\sin 290^{\circ} = -\cos 70^{\circ})
  - (\cos 340^{\circ} = \cos 20^{\circ})
Подставляем: (\sin 20^{\circ} \cdot \sin 20^{\circ} + \cos^{2} 20^{\circ} \cdot \sin^{2} 70^{\circ} + \cos^{2} 70^{\circ} \cdot \cos^{2} 20^{\circ})...