Prove that: cos (2pi/7)+ cos (4pi/7) +cos (8pi/7) = -1/2 . No links

Dear student
We have, cos2π7+cos4π7+cos8π7cos2π7+cos4π7+cos6π7;as cos8π7=cos2π-6π7=cos6π7Multiplying and dividing by 2sinπ7, we have 2sinπ7.cos2π7+2sinπ7.cos4π7+2sinπ7.cos6π72sinπ7=sinπ7-2π7+sinπ7+2π7+sinπ7-4π7+sinπ7+4π7+sinπ7-6π7+sinπ7+6π72sinπ7=sin(-π7)+sin3π7+sin(-3π7)+sin(5π7)+sin(-5π7)+sin(7π7)2sinπ7=-sin(π7)2sinπ7=-12
Hope this clears your doubt 
With regards 

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