Question

Prove the following :
4cosx cos(x+π/3) + cos²(π - 2π/3) = cos3x

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Answer 1

Question :-

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

Answer :-

$\sf \: L.H.S = 4 \cos(x) \times \cos(x + \frac{\pi}{3} ) \times { \cos }^{2} (\pi - \frac{2\pi}{3} )$

$\sf \: = 4 \cos(x) \times (cos \: x \: cos \frac{\pi}{3} - sin \: x \: sin \: \frac{\pi}{3} ) \times (cos \: x \: cos \: \frac{\pi}{3} + \sin \: x \: \sin \frac{\pi}{3} )$

$\sf \: = 4 \cos(x) \times ( \frac{1}{2} \cos \: x \: - \frac{ \sqrt{3} }{2} \sin \: x) \times ( \frac{1}{2} cos \: x + \frac{ \sqrt{3} }{2} \sin \: x)$

$\sf \: = 4 \cos(x) \times ( \frac{1}{4} {cos}^{2} x - \frac{3}{4} {sin}^{2} x)$

$\sf \: = {cos}^{3} x - 3 { \sin }^{2} x \cos(x)$

$\sf \: = {cos}^{3} x - 3(1 - {cos}^{2} x) \cos(x)$

$\sf \: = \cos ^{3} x - 3 \cos(x) + 3cos ^{3} x$

$\sf \: = 4 \cos ^{3} x - 3 \cos(x)$

$\sf \: = \cos(3x)$

$\sf \: = R.H.S$

= hence proved !!

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