How do you prove that #costheta-sinthetasin2theta=costhetacos2theta#?

2 Answers
F. Javier B.
May 26, 2018

#costheta-sinthetasin2theta=costheta-sintheta2sinthetacostheta=#

#=costheta-2sin^2thetacostheta=costheta(1-2sin^2theta)#

But we know that #cos2theta=cos^2theta-sin^2theta=1-sin^2theta-sin^2theta=1-2sin^2theta#

Then, we have #costheta-sinthetasin2theta=costhetacos2theta#

QED

Jim G.
May 26, 2018

#"see explanation"#

Explanation:

#"using the "color(blue)"trigonometric identities"#

#•color(white)(x)sin2theta=2sinthetacostheta#

#•color(white)(x)cos2theta=1-2sin^2theta#

#"consider the left side"#

#costheta-sintheta(2sinthetacostheta)#

#=costheta-2sin^2thetacostheta#

#=costheta(1-2sin^2theta)#

#=costhetacos2theta#

#="right side "rArr"verified"#

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