How do you evaluate #(tan^2((pi)/7) + tan^2((2\pi)/7) + tan^2((3pi)/7))/(cot^2 (pi/7) + cot^2 ((2\pi)/7) + cot^2 ((3\pi)/7))#?

2 Answers
Abhishek K.
Feb 26, 2018

It simplifies to #21/5#.

Explanation:

Here, #tan^2(pi/7)+tan^2((2pi)/7)+tan^2((3pi)/7)#

#=tan^2(pi/7)+tan^2((2pi)/7)+tan^2(pi+(4pi)/7)#

#=tan^2(pi/7)+tan^2((2pi)/7)+tan^2((4pi)/7)#

#=(2sin^2(pi/7))/(2cos^2(pi/7))+(2sin^2((2pi)/7))/(2cos^2((2pi)/7))+(2sin^2((4pi)/7))/(2cos^2((4pi)/7))#

#=(1-cos((2pi)/7))/(1+cos((2pi)/7))+(1-cos((4pi)/7))/(1+cos((4pi)/7))+(1-cos((8pi)/7))/(1+cos((8pi)/7)#

#rarr#Let #pi/7=x#

#=((1-cos2x)(1+cos4x)(1+cos8x)+(1-cos4x)(1+cos2x)(1+cos8x)+(1-cos8x)(1+cos2x)(1+cos4x))/((1+cos2x)(1+cos4x)(1+cos8x))#

#=(1cancel(-cos2x)cancel(+cos4x)+cancel(cos8x)cancel(-cos2x*cos4x)+cancel(cos4x*cos8x)cancel(-cos2x*cos8x)-cos2x*cos4x*cos8x+1+cos2xcancel(-cos4x)+cos8x-cos2x*cos4xcancel(+cos2x*cos8x)-cos4x*cos8x-cos2x*cos4x*cos8x+1cancel(+cos2x)+cos4xcancel(-cos8x)cancel(+cos2x*cos4x)cancel(-cos4x*cos8x)-cos2x*cos8x-cos2x*cos4xcos8x)/(1+cos2x+cos4x+cos8x+cos2x*cos4x+cos4x*cos8x+cos2x*cos8x+cos2x*cos4x*cos8x#

#=(3+cos2x+cos4x+cos8x-(cos2x*cos4x+cos4x*cos8x+cos2x*cos8x)-3(cos2x*cos4x*cos8x))/(1+cos2x+cos4x+cos8x+cos2x*cos4x+cos4x*cos8x+cos2x*cos8x+cos2x*cos4x*cos8x)#

#rarr# Let #a=cos2x+cos4x+cos8x#

#rarra=1/(2sinx)[2cos2xsinx+2cos4xsinx+2cos8xsinx]#

#rarra=1/(2sinx)[sin(2x+x)-sin(2x-x)+sin(4x+x)-sin(4x-x)+sin(8x+x)-sin(8x-x)]#

#rarra=1/(2sinx)[cancel(sin3x)-sinx+sin5xcancel(-sin3x)+sin9x-sin7x]#

#rarra=1/(2sin(pi/7))[sin((5pi)/7)+sin((9pi)/7)-sin((7pi)/7)-sin(pi/7)]#

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

#rarra=1/(2sin(pi/7))[cancel(sin((2pi)/7))cancel(-sin((2pi)/7))-0-sin(pi/7)]#

#a=-1/2#

#rarr# Let #b=cos2x*cos4x+cos4x*cos8x+cos2x*cos8x#

#rarrb=1/2[2cos2x*cos4x+2cos4x*cos8x+2cos2x*cos8x#

#rarrb=1/2[cos6x+cos2x+cos12x+cos4x+cos10x+cos6x]#

#rarrb=1/2[2cos((6pi)/7)+cos((2pi)/7)+cos((12pi)/7)+cos((4pi)/7)+cos((10pi)/7)#

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

#rarrb=2/2[cos((2pi)/7)-cos(pi/7)-cos((3pi)/7)]#

#rarrb=1/(2sin(pi/7))[2cos((2pi)/7)sin(pi/7)-2sin(pi/7)*cos(pi/7)-2cos((3pi/7))*sin(pi/7)]#

#rarrb=1/(2sin(pi/7))[sin((3pi)/7)-sin(pi/7)cancel(-sin((2pi)/7))-sin((4pi)/7)cancel(+sin((2pi)/7))]#

#rarrb=1/(2sin(pi/7))[sin((3pi)/7)-sin(pi/7)-sin(pi-(3pi)/7)]#

#rarrb=1/(2sin(pi/7))[cancel(sin((3pi)/7))-sin(pi/7)-cancel(sin((3pi)/7))]#

#rarrb=-1/2#

#rarr# Let #c=cos2x*cos4xcos8x#

#rarrc=cos((2pi)/7))*cos(pi-(3pi)/7)*cos(pi+pi/7)#

#rarrc=cos(pi/7)*cos((2pi)/7)*cos((3pi)/7)#

#rarrc=1/(2sin(pi/7))[2sin(pi/7)*cos(pi/7)*cos((2pi)/7)*cos((3pi)/7)]#

#rarrc=1/(2*2sin(pi/7))[2sin((2pi)/7)*cos((2pi)/7)*cos((3pi)/7)]#

#rarrc=1/(4sin(pi/7))[sin((4pi)/7)*cos((3pi)/7)]#

#rarrc=1/(4sin(pi/7))[sin(pi-(3pi)/7)*cos((3pi)/7)]#

#rarrc=1/(8sin(pi/7))*sin((6pi)/7)#

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

Now, #tan^2(pi/7)+tan^2((2pi)/7)+tan^2((3pi)/7)#

#=(3+a-b-3c)/(1+a+b+c)#

#=(3-1/2-(-1/2)-3*(1/8))/(1-1/2-1/2+1/8)#

#=(3cancel(-1/2)+cancel(1/2)-3/8)/(cancel(1)cancel(-1)+1/8)#

#=(21/8)/(1/8)=21#

Similarly, #cot^2(pi/7)+cot^2((2pi)/7)+cot^2((3pi)/7)=5#

So, #(tan^2(pi/7)+tan^2((2pi)/7)+tan^2((3pi)/7))/(cot^2(pi/7)+cot^2((2pi)/7)+cot^2((3pi)/7))=21/5#

I'll add some steps for the denominator later.

P dilip_k
Mar 17, 2018

The numerator of the given expression

#=tan^2(pi/7)+tan^2((2pi)/7)+tan^2((3pi)/7)#

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

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

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

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

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

#=2[(3+2cos((2pi)/7)+2cos((4pi)/7)+2cos((6pi)/7)+1/2(cos((6pi)/7)+cos((2pi)/7)+cos((10pi)/7)+cos((2pi)/7)+cos((8pi)/7)+cos((4pi)/7)))/((1+cos((2pi)/7))(1+cos((4pi)/7))(1+cos((6pi) /7)))]-3#

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

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

#=2[(3+3cos((2pi)/7)+3cos((4pi)/7)+3cos((6pi)/7))/(1+cos((2pi)/7)+cos((4pi)/7)+cos((6pi)/7)+1/2(cos((6pi)/7)+cos((2pi)/7)+cos((10pi)/7)+cos((2pi)/7)+cos((8pi)/7)+cos((4pi)/7))+cos((2pi)/7)cos((4pi)/7)cos((6pi)/7))]-3#

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

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

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

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

#=(2xx3/2)/(1/8)-3=24-3=21#

  • Please note

#cos((2pi)/7)+cos((4pi)/7)+cos((6pi)/7)#

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

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

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

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

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

#=-1/2#

Again

#cos((2pi)/7)cos((4pi)/7)cos((6pi)/7)#

#=1/(2sin((2pi)/7))[2sin((2pi)/7)cos((2pi)/7)cos((4pi)/7)cos((6pi)/7)#

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

#=1/(8sin((2pi)/7))[2sin((8pi)/7)cos((6pi)/7)]#

#=1/(8sin((2pi)/7))[2sin(2pi-(6pi)/7)cos((6pi)/7)]#

#=1/(8sin((2pi)/7))[-2sin((6pi)/7)cos((6pi)/7)]#

#=-1/(8sin((2pi)/7))sin((12pi)/7)#

#=-1/(8sin((2pi)/7))sin(2pi-(2pi)/7)#

#=+1/(8sin((2pi)/7))sin((2pi)/7)=1/8#

Again the denominator of the expression

#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#

So whole given expression becomes

#color(magenta)(=(tan^2(pi/7)+tan^2((2pi)/7)+tan^2((3pi)/7))/(cot^2(pi/7)+cot^2((2pi)/7)+cot^2((3pi)/7))=21/5)#

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