I want to show that if ABC is

Jewel Beard

Jewel Beard

Answered question

2022-04-09

I want to show that if ABC is a triangle then
sin2(A2)+sin2(B2)+sin2(C2)=12sin(A2)sin(B2)sin(C2)
Well I eventually got it after much algebra, but I am looking for a shorter solution, or maybe even a geometric one?

Answer & Explanation

libertydragonrbha

libertydragonrbha

Beginner2022-04-10Added 15 answers

Let α=π2A, β=π2B and γ=π2C
Thus, α+β+γ=π and we need to prove that
cos2α+cos2β+cos2γ+2cosαcosβcosγ=1
which is obvious for acute-angled triangle ABC (it's just law of cosines for new triangle).
In the general case we obtain:
cos2α+cos2β+cos2γ+2cosαcosβcosγ=
=cos2α+cos2β+cos2(α+β)2cosαcosβcos(α+β)=
=cos2α+cos2β+cos2αcos2β+sin2αsin2β2sinαsinβcosαcosβ
2cosαcosβ(cosαcosβsinαsinβ)=
=cos2α+cos2βcos2αcos2β+sin2αsin2β=
=cos2αsin2β+cos2β+sin2αsin2β=sin2β+cos2β=1
Done!

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