Вычислить: 1) cos 105 + cos 75; 2) sin 105 - sin 75; 3) cos 11pi/12 + cos 5pi/12; 4) cos 11pi/12 - cos 5pi/12; 5) sin 7pi/12 - sin pi/12; 6) sin 105 + sin 165.

Для решения воспользуемся формулами суммы и разности тригонометрических функций: - $\cos \alpha + \cos \beta = 2 \cos \frac{\alpha + \beta}{2} \cos \frac{\alpha - \beta}{2}$ - $\cos \alpha - \cos \beta = -2 \sin \frac{\alpha + \beta}{2} \sin \frac{\alpha - \beta}{2}$ - $\sin \alpha + \sin \beta = 2 \sin \frac{\alpha + \beta}{2} \cos \frac{\alpha - \beta}{2}$ - $\sin \alpha - \sin \beta = 2 \cos \frac{\alpha + \beta}{2} \sin \frac{\alpha - \beta}{2}$ 1) $\cos 105^{\circ} + \cos 75^{\circ} = 2 \cos \frac{105^{\circ} + 75^{\circ}}{2} \cos \frac{105^{\circ} - 75^{\circ}}{2} = 2 \cos 90^{\circ} \cos 15^{\circ} = 2 \cdot 0 \cdot \cos 15^{\circ} = 0$ 2) $\sin 105^{\circ} - \sin 75^{\circ} = 2 \cos \frac{105^{\circ} + 75^{\circ}}{2} \sin \frac{105^{\circ} - 75^{\circ}}{2} = 2 \cos 90^{\circ} \sin 15^{\circ} = 2 \cdot 0 \cdot \sin 15^{\circ} = 0$ 3) $\cos \frac{11\pi}{12} + \cos \frac{5\pi}{12} = 2 \cos \frac{\frac{16\pi}{12}}{2} \cos \frac{\frac{6\pi}{12}}{2} = 2 \cos \frac{2\pi}{3} \cos \frac{\pi}{4} = 2 \cdot (-\frac{1}{2}) \cdot \frac{\sqrt{2}}{2} = -\frac{\sqrt{2}}{2}$ 4) $\cos \frac{11\pi}{12} - \cos \frac{5\pi}{12} = -2 \sin \frac{\frac{16\pi}{12}}{2} \sin \frac{\frac{6\pi}{12}}{2} = -2 \sin \frac{2\pi}{3} \sin \frac{\pi}{4} = -2 \cdot \frac{\sqrt{3}}{2} \cdot \frac{\sqrt{2}}{2} = -\frac{\sqrt{6}}{2}$ 5) $\sin \frac{7\pi}{12} - \sin \frac{\pi}{12} = 2 \cos \frac{\frac{8\pi}{12}}{2} \sin \frac{\frac{6\pi}{12}}{2} = 2 \cos \frac{\pi}{3} \sin \frac{\pi}{4} = 2 \cdot \frac{1}{2} \cdot \frac{\sqrt{2}}{2} = \frac{\sqrt{2}}{2}$ 6) $\sin 105^{\circ} + \sin 165^{\circ} = 2 \sin \frac{105^{\circ} + 165^{\circ}}{2} \cos \frac{105^{\circ} - 165^{\circ}}{2} = 2 \sin 135^{\circ} \cos(-30^{\circ}) = 2 \cdot \frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} = \frac{\sqrt{6}}{2}$