若α+β=2π/3,则(1-根号3tanα)(1-根号3tanβ)

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若α+β=2π/3,则(1-根号3tanα)(1-根号3tanβ)若α+β=2π/3,则(1-根号3tanα)(1-根号3tanβ)若α+β=2π/3,则(1-根号3tanα)(1-根号3tanβ)t

若α+β=2π/3,则(1-根号3tanα)(1-根号3tanβ)
若α+β=2π/3,则(1-根号3tanα)(1-根号3tanβ)

若α+β=2π/3,则(1-根号3tanα)(1-根号3tanβ)
tan(α+β)=(tanα+tanβ)/(1-tanαtanβ)=-根号3
所以tanα+tanβ=-根号3(1-tanαtanβ)
原式=1-根号3(tanα+tanβ)+3tanαtanβ
=1+3=4

利用tan的和角公式:tan(x+y)=(tanx+tany)/(1-tanxtany) 即有
tanx+tany=tan(x+y)(1-tanxtany),注意到tan (α+β)=tan2π/3=-根号3,
所以
(1-根号3tanα)(1-根号3tanβ)
=1-根号3tanα-根号3tanβ+3tanαtanβ
=1-根号3(tanα+tanβ)+3...

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利用tan的和角公式:tan(x+y)=(tanx+tany)/(1-tanxtany) 即有
tanx+tany=tan(x+y)(1-tanxtany),注意到tan (α+β)=tan2π/3=-根号3,
所以
(1-根号3tanα)(1-根号3tanβ)
=1-根号3tanα-根号3tanβ+3tanαtanβ
=1-根号3(tanα+tanβ)+3tanαtanβ
=1-根号3*tan(α+β)*(1-tanαtanβ)+3tanαtanβ
=1+3(1-tanαtanβ)+3tanαtanβ
=4

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