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Answers
Answer:
LHS is solved
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Answer:
Cosα + Cosβ + Cosγ + Cos(α + β+ γ) = 4Cos(α+β/2)Cos(β+γ/2)Cos(α+γ/2)
Step-by-step explanation:
Cosα + Cosβ + Cosγ + Cos(α + β+ γ) = 4Cos(α+β/2)Cos(β+γ/2)Cos(α+γ/2)
Using CosA + CosB = 2Cos((A + B)/2)Cos((A - B)/2)
LHS = Cosα +Cosβ + Cosγ + Cos(α + β+ γ)
= 2Cos((α + β)/2)Cos((α - β)/2) + 2 Cos((α + β+ 2γ)/2)Cos((-α - β)/2)
Using Cos(-A) = CosA
= 2Cos((α + β)/2)Cos((α - β)/2) + 2 Cos((α + β+ 2γ)/2)Cos((α + β)/2)
= 2Cos((α + β)/2) (Cos((α - β)/2) + Cos((α + β+ 2γ)/2))
= 2Cos((α + β)/2) ( 2 Cos((α + γ)/2)Cos((-β - γ)/2)
= 2Cos((α + β)/2) ( 2 Cos((α + γ)/2) (Cos((β + γ)/2))
= 4 Cos((α + β)/2)Cos((β + γ)/2)Cos((α + γ)/2)
= RHS
QED
Proved
Cosα + Cosβ + Cosγ + Cos(α + β+ γ) = 4Cos(α+β/2)Cos(β+γ/2)Cos(α+γ/2)
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