Tan(pi/12)tan(5pi/12)tan(7pi/12)tan(11pi/12)=?

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Answer 1 · 2018-06-20T10:20:33

#Tan(pi/12)tan((5pi)/12)tan((7pi)/12)tan((11pi)/12)=1#

Let ,
#X=tan(pi/12)color(red) (tan((5pi)/12)tan((7pi)/12))tan((11pi)/12)#

We know that,

#(i)tan(pi/2-theta)=cottheta # ,

putting #totheta=pi/12#

#=>tan(pi/2-pi/12)=cot(pi/12)#

#=>tan((6pi-pi)/12)=cot(pi/12)#

#=>color(red)(tan((5pi)/12)=cot(pi/12)...to(1)#

#(ii)tan((3pi)/2-theta)=cottheta # ,

putting #totheta=(11pi)/12#

#=>tan((3pi)/2-(11pi)/12)=cot((11pi)/12)#

#=>tan((18pi-11pi)/12)=cot((11pi)/12)#

#=>color(red)(tan((7pi)/12)=cot((11pi)/12)...to(2)#

Using #(1) and (2) and color(blue)(tantheta*cottheta=1#

#X=tan(pi/12)color(red)(tan((5pi)/12)tan((7pi)/12))tan((11pi)/12)#

#X=tan(pi/12)color(red)(cot((pi)/12)cot((11pi)/12))tan((11pi)/12)#

#X=color(blue)((1)*(1)#

#X=1#