If α,β and γ are three distinct real values such that sin⁡α+sin⁡β+sin⁡γsin⁡(α+β+γ)=cos⁡α+cos⁡β+cos⁡γcos⁡(α+β+γ)=2 then cos⁡(α+β)+cos⁡(β+γ)+cos⁡(γ+α)=___

put , a=cis α,b=cisβ, c=cisγ⇒a+b+c=2abc⇒∑ 1ab=2⇒∑ cosα+β=2sin⁡α+sin⁡β+sin⁡γsin⁡(α+β+γ)=cos⁡α+cos⁡β+cos⁡γcos⁡(α+β+γ)sin⁡α+β+γsinα+sinβ+sinγ+cosα+β+γcosα+cosβ+cosγsin⁡(α+β+γ)sin⁡α+β+γ+cosα+β+γcosα+β+γ∴ab=cd=xa+ycxb+yd (By ratio and proportion)∴cosα+β+cosβ+γ+cosγ+α1=2

If α,β and γ are three distinct real values such that sin⁡α+sin⁡β+sin⁡γsin⁡(α+β+γ)=cos⁡α+cos⁡β+cos⁡γcos⁡(α+β+γ)=2 then cos⁡(α+β)+cos⁡(β+γ)+cos⁡(γ+α)=___

put , a=cis α,b=cisβ, c=cisγ⇒a+b+c=2abc⇒∑ 1ab=2⇒∑ cosα+β=2sin⁡α+sin⁡β+sin⁡γsin⁡(α+β+γ)=cos⁡α+cos⁡β+cos⁡γcos⁡(α+β+γ)sin⁡α+β+γsinα+sinβ+sinγ+cosα+β+γcosα+cosβ+cosγsin⁡(α+β+γ)sin⁡α+β+γ+cosα+β+γcosα+β+γ∴ab=cd=xa+ycxb+yd (By ratio and proportion)∴cosα+β+cosβ+γ+cosγ+α1=2

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If $α, β$ and $γ$ are three distinct real values such that $\frac{\sin α + \sin β + \sin γ}{\sin (α + β + γ)} = \frac{\cos α + \cos β + \cos γ}{\cos (α + β + γ)} = 2$ then $\cos (α + β) + \cos (β + γ) + \cos (γ + α) =$ ___

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Detailed Solution

put , a=cis $α$ ,b=cis $β$ , c=cis $γ$
$\Rightarrow a + b + c = 2 abc \Rightarrow \sum_{} \frac{1}{ab} = 2$
$\Rightarrow \sum_{} \cos (α + β) = 2$
$\frac{\sin α + \sin β + \sin γ}{\sin (α + β + γ)} = \frac{\cos α + \cos β + \cos γ}{\cos (α + β + γ)}$
$\frac{\sin (α + β + γ) (sinα + \sin β + \sin γ) + \cos (α + β + γ) (cosα + \cos β + \cos γ)}{\sin (α + β + γ) \sin (α + β + γ) + \cos (α + β + γ) \cos (α + β + γ)}$
$(∴ \frac{a}{b} = \frac{c}{d} = \frac{xa + yc}{xb + yd})$ (By ratio and proportion)
$∴ \frac{\cos (α + β) + \cos (β + γ) + \cos (γ + α)}{1} = 2$