1.

If cos(α – β) + cos(β – γ) + cos(γ – α) = -3/2, then prove that cos α + cos β + cos γ = sin α + sin β + sin γ

Answer»

Given cos(α – β) + cos(β – γ) + cos(γ – α) = -3/2

2 cos (α – β) + 2cos (β – γ) + 2cos (γ – α) = -3 

2cos(α – β) + 2cos(β – γ) + 2cos (γ – α) + 3 = 0 

[2 cos α cos β + 2 sin α sin β] + [2 cos β cos γ + 2 sin β sin γ] + [2 cos γ cos α + sin γ sin α] + 3 = 0 

= [2 cos α cos β + 2 cos β cos γ + 2 cos γ cos α] + [2 sin α sin β + 2 sin β sin γ + 2 sin γ sin α] + (sin2 α + cos2 α) + (sin2 β + cos2 β) + (sin2 γ + cos2 γ) = 0 

⇒ (cos2 α + cos2 β + cos2 γ + 2 cos α cos β + 2 cos β cos γ + 2 cos γ cos α) + (sin2 α + sin2 β) + (sin2 γ + 2 sin α sin β + 2 sin β sin γ + 2 sin γ sin α) = 0 

(cos α + cos β + cos γ)2 + (sin α + sin β + sin γ)2 = 0 

=(cos α + cos β + cos γ) = 0 and sin α + sin β + sin γ = 0 

Hence proved


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