cos(2π/7)+cos(4π/7)+cos(6π/7)= -1/2.证明[cos(2π/7)]^2+[cos(4π/7)]^2+[cos(6π/7)]^2=5/4

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cos(2π/7)+cos(4π/7)+cos(6π/7)=-1/2.证明[cos(2π/7)]^2+[cos(4π/7)]^2+[cos(6π/7)]^2=5/4cos(2π/7)+cos(4π/7

cos(2π/7)+cos(4π/7)+cos(6π/7)= -1/2.证明[cos(2π/7)]^2+[cos(4π/7)]^2+[cos(6π/7)]^2=5/4
cos(2π/7)+cos(4π/7)+cos(6π/7)= -1/2.
证明[cos(2π/7)]^2+[cos(4π/7)]^2+[cos(6π/7)]^2=5/4

cos(2π/7)+cos(4π/7)+cos(6π/7)= -1/2.证明[cos(2π/7)]^2+[cos(4π/7)]^2+[cos(6π/7)]^2=5/4
cos(2π/7)+cos(4π/7)+cos(6π/7)= -1/2
[cos(2π/7)+cos(4π/7)+cos(6π/7)]^2= 1/4
[cos(2π/7)]^2+[cos(4π/7)]^2+[cos(6π/7)]^2+2cos(2π/7)cos(4π/7)+2cos(2π/7)cos(6π/7)+2cos(4π/7)cos(6π/7)=1/4

[cos(2π/7)]^2+[cos(4π/7)]^2+[cos(6π/7)]^2 = 1/4 - 2[cos(2π/7)cos(4π/7)+cos(4π/7)cos(6π/7)+cos(2π/7)cos(6π/7)]
应用积化和差公式
cos(2π/7)cos(4π/7)+cos(4π/7)cos(6π/7)+cos(2π/7)cos(6π/7)
=(cos6π/7+cos2π/7)/2+(cos10π/7+cos2π/7)/2+(cos8π/7+cos4π/7)/2
=cos(2π/7)+cos(4π/7)+cos(6π/7)=-1/2
所以
[cos(2π/7)]^2+[cos(4π/7)]^2+[cos(6π/7)]^2 = 1/4 - 2*(-1/2) = 5/4