The value of ((2π)/(7)) + ((4π)/(7)) ++ ((6π)/(7))is equal to

Given, $cos\dfrac{2\pi}{7} + cos\dfrac{4\pi}{7} + cos\dfrac{6\pi}{7}$

$= \dfrac{1}{2\sin \left(\dfrac{π}{7}\right)}\left\{2\cos \left(\dfrac{2\pi}{7}\right)\sin \left(\dfrac{π}{7}\right) + 2\cos \left(\dfrac{4\pi}{7}\right)\sin \left(\dfrac{π}{7}\right) + 2\cos \left(\dfrac{6\pi}{7}\right)\sin \left(\dfrac{π}{7}\right)\right\}$

$= \dfrac{1}{2sin\dfrac{π}{7}}\left\{sin\dfrac{3\pi}{7} - sin\dfrac{π}{7} + sin\dfrac{5\pi}{7} - sin\dfrac{3\pi}{7} + sin\dfrac{7\pi}{7} - sin\dfrac{5\pi}{7}\right\}$

Since, 2cosAsinB = \sin \left(A + B\right) - \sin \left(A - B\right)$</p><p>=\dfrac{1}{2sin\dfrac{π}{7}} \left\{- sin\dfrac{π}{7} +sin \pi\right\}$$</p><p>=\dfrac{- 1}{2}, \left(∵ sin \pi =0\right)$