Представьте в виде произведения: а) cos π/10 - cos π/20; б) cos 11π/12 + cos 3π/4; в) cos π/5 - cos π/11; г) cos 3π/8 + cos 5π/4.

Для решения воспользуемся формулами суммы и разности косинусов: $\cos \alpha - \cos \beta = -2 \sin \frac{\alpha + \beta}{2} \sin \frac{\alpha - \beta}{2}$ $\cos \alpha + \cos \beta = 2 \cos \frac{\alpha + \beta}{2} \cos \frac{\alpha - \beta}{2}$ а) $\cos \frac{\pi}{10} - \cos \frac{\pi}{20} = -2 \sin \frac{\frac{\pi}{10} + \frac{\pi}{20}}{2} \sin \frac{\frac{\pi}{10} - \frac{\pi}{20}}{2} = -2 \sin \frac{\frac{3\pi}{20}}{2} \sin \frac{\frac{\pi}{20}}{2} = -2 \sin \frac{3\pi}{40} \sin \frac{\pi}{40}$ б) $\cos \frac{11\pi}{12} + \cos \frac{3\pi}{4} = 2 \cos \frac{\frac{11\pi}{12} + \frac{9\pi}{12}}{2} \cos \frac{\frac{11\pi}{12} - \frac{9\pi}{12}}{2} = 2 \cos \frac{\frac{20\pi}{12}}{2} \cos \frac{\frac{2\pi}{12}}{2} = 2 \cos \frac{5\pi}{6} \cos \frac{\pi}{12}$ в) $\cos \frac{\pi}{5} - \cos \frac{\pi}{11} = -2 \sin \frac{\frac{\pi}{5} + \frac{\pi}{11}}{2} \sin \frac{\frac{\pi}{5} - \frac{\pi}{11}}{2} = -2 \sin \frac{\frac{16\pi}{55}}{2} \sin \frac{\frac{6\pi}{55}}{2} = -2 \sin \frac{8\pi}{55} \sin \frac{3\pi}{55}$ г) $\cos \frac{3\pi}{8} + \cos \frac{5\pi}{4} = 2 \cos \frac{\frac{3\pi}{8} + \frac{10\pi}{8}}{2} \cos \frac{\frac{3\pi}{8} - \frac{10\pi}{8}}{2} = 2 \cos \frac{13\pi}{16} \cos \left(-\frac{7\pi}{16}\right) = 2 \cos \frac{13\pi}{16} \cos \frac{7\pi}{16}$