Saturday, 22 March 2014

trigonometry - Prove that cosfrac2pi7+cosfrac4pi7+cosfrac8pi7=frac12





Prove that
cos2π7+cos4π7+cos8π7=12




My attempt



LHS=cos2π7+cos4π7+cos8π7=2cos4π7cosπ7+2cos24π71=2cos4π7(cosπ7cos4π7)1


Now, please help me to complete the proof.


Answer



cos(2π/7)+cos(4π/7)+cos(8π/7)



= cos(2π/7)+cos(4π/7)+cos(6π/7) (angles add to give 2π, thus one is 2π minus the other)




At this point, we'll make an observation



cos(2π/7)sin(π/7) = sin(3π/7)sin(π/7)2 ..... (A)



cos(4π/7)sin(π/7) = sin(5π/7)sin(3π/7)2 ..... (B)



cos(6π/7)sin(π/7) = sin(7π/7)sin(5π/7)2 ..... (C)



Now, add (A), (B) and (C) to get




sin(π/7)(cos(2π/7)+cos(4π/7)+cos(6π/7)) = sin(7π/7)sin(π/7)2 = -sin(π/7)/2



The sin(π/7) cancels out from both sides to give you your answer.


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