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Question Number 168297 by MdNafiz last updated on 07/Apr/22

prove that........  cos ((2Π)/7)+cos ((4Π)/7)+cos ((8Π)/7)=−(1/2)

provethat........cos2Π7+cos4Π7+cos8Π7=12

Answered by som(math1967) last updated on 07/Apr/22

let (π/7)=θ    ∴cos2θ+cos4θ+cos8θ  =((2sin2θcos2θ+2sin2θcos4θ+2sin2θcos8θ)/(2sin2θ))  =((sin4θ+sin6θ−sin2θ+sin10θ−sin6θ)/(2sin2θ))  =((sin4θ+sin10θ−sin2θ)/(2sin2θ))  =((2sin7θcos3θ−sin2θ)/(2sin2θ))  =((0−sin2θ)/(2sin2θ))=−(1/2) ★  ★sin7θ=sinπ=0

letπ7=θcos2θ+cos4θ+cos8θ=2sin2θcos2θ+2sin2θcos4θ+2sin2θcos8θ2sin2θ=sin4θ+sin6θsin2θ+sin10θsin6θ2sin2θ=sin4θ+sin10θsin2θ2sin2θ=2sin7θcos3θsin2θ2sin2θ=0sin2θ2sin2θ=12sin7θ=sinπ=0

Commented by peter frank last updated on 08/Apr/22

good

good

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