Prove: sin ⁡ α + sin ⁡ β + sin ⁡ γ = 4 cos ⁡ α 2 cos ⁡ β 2 cos ⁡ γ 2 when α + β + γ = π

Question

Prove:   sin  ⁡  α  +  sin  ⁡  β  +  sin  ⁡  γ  =  4  cos  ⁡            α      2        cos  ⁡            β      2        cos  ⁡            γ      2       when   α  +  β  +  γ  =  π

Lana Schwartz · Accepted Answer

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  ⁡  δto 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

Prove: sin ⁡ α + sin ⁡ β<!-

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