Монгол Бодлогын Сан

Заавар:

Бодолт: $$\begin{align*} \cos\dfrac{\pi}{7}\cdot\cos\dfrac{4\pi}{7}\cdot\cos\dfrac{5\pi}{7}&=-\cos\dfrac{\pi}{7}\cdot\cos\dfrac{2\pi}{7}\cdot\cos\dfrac{4\pi}{7} & & \color{red}{\leftarrow \cos\dfrac{5\pi}{7}=-\dfrac{2\pi}{7}}\\ &=-\dfrac{8\sin\dfrac{\pi}{7}\cdot\cos\dfrac{\pi}{7}\cdot\cos\dfrac{2\pi}{7}\cdot\cos\dfrac{4\pi}{7}}{8\sin\dfrac{\pi}{7}}\\ &=-\dfrac{4\sin\dfrac{2\pi}{7}\cdot\cos\dfrac{2\pi}{7}\cdot\cos\dfrac{4\pi}{7}}{8\sin\dfrac{\pi}{7}} & & \color{red}{\leftarrow 2\sin\alpha\cos\alpha=\sin2\alpha}\\ &=-\dfrac{2\sin\dfrac{4\pi}{7}\cdot\cos\dfrac{4\pi}{7}}{8\sin\dfrac{\pi}{7}} & & \color{red}{\leftarrow 2\sin\alpha\cos\alpha=\sin2\alpha}\\ &=-\dfrac{\sin\dfrac{8\pi}{7}}{8\sin\dfrac{\pi}{7}}=1 & & \color{red}{\leftarrow \sin\dfrac{8\pi}{7}=-\sin\dfrac{\pi}{7}} \end{align*}$$