Question

Prove that: Cos x + Cos(x+ 2pi/3) + Cos (x + 4pi/3) =0

Answer 1

$\boxed {\pink { cos A + Cos B = 2cos \Big(\frac{A+B}{2}\Big) cos \Big(\frac{A-B}{2}\Big)}}$

$LHS = cos x + cos \Big( x +\frac{2\pi}{3}\Big) +cos \Big( x +\frac{4\pi}{3}\Big)$

$= cosx + 2 cos \Big ( \frac{x+\frac{2\pi}{3} + x + \frac{4\pi}{3}}{2}\Big) cos \Big ( \frac{x+\frac{2\pi}{3} - \big( x + \frac{4\pi}{3}\big)}{2}\Big)$

$= cos x + 2 cos \Big (\frac{2x + \frac{2\pi +4\pi}{3}}{2}\Big) cos \Big (\frac{\frac{2\pi -4\pi}{3}}{2}\Big)$

$= cosx + 2 cos \frac{(2x + 2\pi)}{2} cos \frac{(-\pi)}{3}$

$= cosx + 2 cos ( \pi + x ) cos\frac{\pi}{3}$

$= cosx + 2 (-cos x ) \times \frac{1}{2}$

$= cos x - cos x \\= 0 \\= RHS$

$Hence\: Proved$

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Answer 2

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