How do you prove $cos 3 theta = 4 cos^3 theta - 3 cos theta$ ?

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Answer 1 · 2016-06-06T17:18:39

Proof is given below.

$cos3theta=cos(2theta+theta)$

$=cos2thetacostheta-sin2thetasintheta$

$=(cos^2theta-sin^2theta)costheta-2sinthetacosthetasintheta$

$=cos^3theta-sin^2costheta-2sin^2thetacostheta$

$=costheta(cos^2theta-sin^2theta-2sin^2theta)$

$=costheta(cos^2theta-3sin^2theta)$

$=cos^3theta-3sin^2thetacostheta$

$=cos^3theta-3(1-cos^2theta)costheta$

$=cos^3theta-3costheta+3cos^3theta$

$=4cos^3theta-3costheta$

How do you prove cos 3 theta = 4 cos^3 theta - 3 cos thetacos 3 theta = 4 cos^3 theta - 3 cos theta?