\(cos\dfrac{\pi}{7}-cos\dfrac{2\pi}{7}+cos\dfrac{3\pi}{7}\)
\(=-2.sin\dfrac{3\pi}{14}.sin\dfrac{-\pi}{14}+cos\left(\dfrac{\pi}{2}-\dfrac{\pi}{14}\right)\)
\(=2.sin\dfrac{3\pi}{14}.sin\dfrac{\pi}{14}+sin\dfrac{\pi}{14}\)
\(=sin\dfrac{\pi}{14}\left(1+2sin\dfrac{3\pi}{14}\right)\)\(=sin\dfrac{\pi}{14}\left(1+2.sin\left(\dfrac{\pi}{2}-\dfrac{2\pi}{7}\right)\right)\)\(=sin\dfrac{\pi}{14}\left(1+2.cos\dfrac{2\pi}{7}\right)\)
\(=sin\dfrac{\pi}{14}\left(2cos^2\dfrac{\pi}{7}+cos\dfrac{2\pi}{7}\right)\)\(=\dfrac{sin\dfrac{\pi}{7}}{2.cos\dfrac{\pi}{14}}\left(2.cos^2\dfrac{\pi}{7}+cos\dfrac{2\pi}{7}\right)\)
\(=\dfrac{1}{2}.\dfrac{1}{cos\dfrac{\pi}{14}}\left(2.sin\dfrac{\pi}{7}.cos^2\dfrac{\pi}{7}+sin\dfrac{\pi}{7}.cos\dfrac{2\pi}{7}\right)\)
\(=\dfrac{1}{2}.\dfrac{1}{cos\dfrac{\pi}{14}}\left(sin\dfrac{2\pi}{7}.cos\dfrac{\pi}{7}+cos\dfrac{2\pi}{7}.sin\dfrac{\pi}{2}\right)\)
\(=\dfrac{1}{2}.\dfrac{1}{cos\dfrac{\pi}{14}}.sin\dfrac{3\pi}{7}\)\(=\dfrac{1}{2}.\dfrac{1}{cos\dfrac{\pi}{14}}.cos\dfrac{\pi}{14}\)
\(=\dfrac{1}{2}\)(đpcm)
