a,b,c是公差为pi/3的等差数列tanatanb+tanbtanc+tanctana=?

Question

a,b,c是公差为pi/3的等差数列 tanatanb+tanbtanc+tanctana=?

Answer 1

a,b,c是公差为π/3的等差数列
则c-b=π/3 b-a=π/3 a-c=-2π/3
tan(c-b)=(tanc-tanb)/(1+tanbtanc)
tanc-tanb=tanπ/3(1+tanbtanc)
√3+√3tanbtanc=tanc-tanb①
tan(b-a)=(tanb-tana)/(1+tanatanb)
tanb-tana=tanπ/3(1+tanatanb)
√3+√3tanatanb=tanb-tana②
tan(a-c)=(tana-tanc)/(1+tanctana)
(tana-tanc)=tan(-2π/3)(1+tanctana)
√3+√3tanctana=tana-tanc③
①+②+③
3√3+√3(tanatanb+tanbtanc+tanctana)=0
tanatanb+tanbtanc+tanctana=-3