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Ximena Skinner 2022-07-08 Answered
Prove: sin α + sin β + sin γ = 4 cos α 2 cos β 2 cos γ 2 when α + β + γ = π
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Answers (1)

Lana Schwartz
Answered 2022-07-09 Author has 8 answers
You may go this way:
cos γ 2 = cos π α β 2 = sin α + β 2 = sin α 2 cos β 2 + cos α 2 sin β 2
so the right hand side becomes
4 cos α 2 cos β 2 sin α 2 cos β 2 + 4 cos α 2 cos β 2 cos α 2 sin β 2
Recalling the duplication formula for the sine we get
2 sin α cos 2 β 2 + 2 sin β cos 2 α 2
and we can recall
2 cos 2 δ 2 = 1 + cos δ
to get
sin α + sin α cos β + sin β + sin β cos α = sin α + sin β + sin ( α + β ) = sin α + sin β + sin γ

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