How do you prove that #cot^2(pi/7)+cot^2((2pi)/7)+cot^2((3pi)/7)=5#?

1 Answer
P dilip_k
Mar 16, 2018

#cot^2(pi/7)+cot^2((2pi)/7)+cot^2((3pi)/7)#

#=csc^2(pi/7)+csc^2((2pi)/7)+csc^2((3pi)/7)-3#

#=1/sin^2(pi/7)+1/sin^2((2pi)/7)+1/sin^2((3pi)/7)-3#

#=2/(1-cos((2pi)/7))+2/(1-cos((4pi)/7))+2/(1-cos((6pi)/7))-3#

#=2/(1+cos((5pi)/7))+2/(1+cos((3pi)/7))+2/(1+cos(pi/7))-3#

#=2[((1+cos((3pi)/7))(1+cos(pi/7))+(1+cos((5pi)/7))(1+cos(pi/7)+(1+cos((5pi)/7))(1+cos((3pi)/7))))/((1+cos((5pi)/7))(1+cos((3pi)/7))(1+cos(pi/7)))]-3#

#=2[(3+2cos((3pi)/7)+2cos(pi/7)+2cos((5pi)/7)+cos(pi/7)cos((3pi)/7)+cos((3pi)/7)cos((5pi)/7)+cos((5pi)/7)cos(pi/7))/((1+cos((5pi)/7))(1+cos((3pi)/7))(1+cos(pi/7)))]-3#

#=2[(3+2cos((3pi)/7)+2cos(pi/7)+2cos((5pi)/7)+1/2(cos((4pi)/7)+cos((2pi)/7)+cos((8pi)/7)+cos((2pi)/7)+cos((6pi)/7)+cos((4pi)/7)))/((1+cos((5pi)/7))(1+cos((3pi)/7))(1+cos(pi/7)))]-3#

#=2[(3+2cos((3pi)/7)+2cos(pi/7)+2cos((5pi)/7)+1/2(-cos((3pi)/7)-cos((5pi)/7)-cos(pi/7)-cos((5pi)/7)-cos(pi/7)-cos((3pi)/7)))/((1+cos((5pi)/7))(1+cos((3pi)/7))(1+cos(pi/7)))]-3#

#=2[(3+cos((3pi)/7)+cos(pi/7)+cos((5pi)/7))/((1+cos((5pi)/7))(1+cos((3pi)/7))(1+cos(pi/7)))]-3#

#=2[(3+cos((3pi)/7)+cos(pi/7)+cos((5pi)/7))/(1+cos((3pi)/7)+cos(pi/7)+cos((5pi)/7)+cos(pi/7)cos((3pi)/7)+cos((3pi)/7)cos((5pi)/7)+cos((5pi)/7)cos(pi/7)+cos((5pi)/7)cos((3pi)/7)cos(pi/7))]-3#

#=2[(3+cos((3pi)/7)+cos(pi/7)+cos((5pi)/7))/((1+cos((3pi)/7)+cos(pi/7)+cos((5pi)/7)+1/2(-cos((3pi)/7)-cos((5pi)/7)-cos(pi/7)-cos((5pi)/7)-cos(pi/7)-cos((3pi)/7))+cos((5pi)/7)cos((3pi)/7))cos(pi/7))]-3#

#=color(red)(2[(3+cos((3pi)/7)+cos(pi/7)+cos((5pi)/7))/(1+cos((5pi)/7)cos((3pi)/7)cos(pi/7))]-3)#

#=2((3+1/2)/(1-1/8))-3# [ please see the note below ]

#=(2xx7/2)/(7/8)-3=5#

  • Please note

#cos((3pi)/7)+cos(pi/7)+cos((5pi)/7))#

#=1/(2sin(pi/7))[2sin(pi/7)cos(pi/7)+2sin((3pi)/7)cos(pi/7)+2sin(pi/7)cos((5pi)/7))]#

#=1/(2sin(pi/7))[sin((2pi)/7)+sin((4pi)/7)-sin((2pi)/7)+sin((6pi)/7)-sin((4pi)/7)]#

#=1/(2sin(pi/7))[sin(pi-pi/7)]#
#=1/(2sin(pi/7))*sin(pi/7)#

#=1/2#

Again

#cos((5pi)/7)cos((3pi)/7)cos(pi/7)#

#=1/(2sin(pi/7))[2sin(pi/7)cos(pi-(2pi)/7)cos(pi-(4pi)/7)cos(pi/7)]#

#=1/(4sin(pi/7))[2sin((2pi)/7)cos((2pi)/7)cos((4pi)/7)]#

#=1/(8sin(pi/7))[2sin((4pi)/7)cos((4pi)/7)]#

#=1/(8sin(pi/7))sin((8pi)/7)#

#=1/(8sin(pi/7))sin(pi+pi/7)#

#=-1/(8sin(pi/7))sin(pi/7)=-1/8#

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