若tan(α/2)=[tan(β/2)]^3,tanβ=2tanφ,证明α+β=2kπ+2φ(k∈Z)

若tan(α/2)=[tan(β/2)]^3,tanβ=2tanφ,证明α+β=2kπ+2φ(k∈Z)
若tan(α/2)=[tan(β/2)]^3,tanβ=2tanφ,证明α+β=2kπ+2φ(k∈Z)

若tan(α/2)=[tan(β/2)]^3,tanβ=2tanφ,证明α+β=2kπ+2φ(k∈Z)
tan(a/2)=(tan(b/2))^3
tan(a/2+b/2)=[tan(a/2)+tan(b/2)]/[1-tan(a/2)tan(b/2)]
=(tanb/2)(1+tan(b/2)^2)/(1-(tanb/2)^4)
=tan(b/2)/[1-(tanb/2)^2]
=tanb/2=tanφ
a/2+b/2=φ+kπ