Prove that
cos2π7+cos4π7+cos8π7=−12
My attempt
LHS=cos2π7+cos4π7+cos8π7=−2cos4π7cosπ7+2cos24π7−1=−2cos4π7(cosπ7−cos4π7)−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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