Tính sin 2pi/7 + sin 4pi/7 + sin 6pi/7. 1/2 .cot pi/4

* Hướng dẫn giải

Đáp án A

$$\begin{gathered} \sin \frac{\pi }{7}\left(\sin \frac{2\pi }{7}+\sin \frac{4\pi }{7}+\sin \frac{6\pi }{7}\right) \\ =\sin \frac{\pi }{7}\sin \frac{2\pi }{7}+\sin \frac{\pi }{7}\sin \frac{4\pi }{7}+\sin \frac{\pi }{7}\sin \frac{6\pi }{7} \\ =\frac{1}{2}\left(cos\frac{\pi }{7}-cos\frac{3\pi }{7}\right)+\frac{1}{2}\left(cos\frac{3\pi }{7}-cos\frac{5\pi }{7}\right)+\frac{1}{2}\left(cos\frac{5\pi }{7}-cos\frac{7\pi }{7}\right) \\ =\frac{1}{2}cos\frac{\pi }{7}+\frac{1}{2}=co{s}^2\frac{\pi }{14} \\ \sin \frac{\pi }{7}=2\sin \frac{\pi }{14}cos\frac{\pi }{14} \\ \Rightarrow \sin \frac{2\pi }{7}+\sin \frac{4\pi }{7}+\sin \frac{6\pi }{7}=\frac{1}{2}\cot \frac{\pi }{14} \end{gathered}$$