Монгол Бодлогын Сан

Заавар: $A^2$-ийг зэрэг буурах томьёо ашиглан бод.

Бодолт: $$\begin{align*} A^2&{}=\sin^2\dfrac{\pi}{7}\cdot\sin^2\dfrac{2\pi}{7}\cdot\sin^2\dfrac{3\pi}{7}\\ &{}=\dfrac{1-\cos\dfrac{2\pi}{7}}{2}\cdot\dfrac{1-\cos\dfrac{4\pi}{7}}{2}\cdot\dfrac{1-\cos\dfrac{6\pi}{7}}{2}\\ &{}=\dfrac18\left(1-\cos\dfrac{2\pi}{7}\right)\left(1-\cos\dfrac{4\pi}{7}\right)\left(1-\cos\dfrac{6\pi}{7}\right)\\ &{}=\dfrac18\left(1-\cos\dfrac{2\pi}{7}-\cos\dfrac{4\pi}{7}-\cos\dfrac{6\pi}{7}+\cos\dfrac{2\pi}{7}\cdot\cos\dfrac{4\pi}{7}\right.\\ &{}+\left.\cos\dfrac{2\pi}{7}\cdot\cos\dfrac{6\pi}{7}+\cos\dfrac{4\pi}{7}\cdot\cos\dfrac{6\pi}{7}-\cos\dfrac{\pi}{7}\cdot\cos\dfrac{4\pi}{7}\cdot\cos\dfrac{6\pi}{7}\right) \end{align*}$$