If α,β,γ∈(0,π2), then sin(α+β+γ)sinα+sinβ+sinγ is

sinα+sinβ+sinγ−sin(α+β+γ) =2sinα+β2cosα−β2−2sinα+β2cos(α+β2+γ) =2sinα+β2[2sinα+γ2sinγ+β2] 4sinα+β2sinα+γ2sinγ+β2>0∵0 sin(α+β+γ) ⇒sin(α+β+γ)sinα+sinβ+sinγ<1

If α,β,γ∈(0,π2), then sin(α+β+γ)sinα+sinβ+sinγ is

sinα+sinβ+sinγ−sin(α+β+γ) =2sinα+β2cosα−β2−2sinα+β2cos(α+β2+γ) =2sinα+β2[2sinα+γ2sinγ+β2] 4sinα+β2sinα+γ2sinγ+β2>0∵0 sin(α+β+γ) ⇒sin(α+β+γ)sinα+sinβ+sinγ<1

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If $α, β, γ \in (0, \frac{π}{2}),$ then $\frac{\sin (α + β + γ)}{\sin α + \sin β + \sin γ}$ is

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$< 1$

$> 1$

$= 1$

$< 0$

answer is A.

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Detailed Solution

$\sin α + \sin β + \sin γ - \sin (α + β + γ)$
$= 2 \sin \frac{α + β}{2} \cos \frac{α - β}{2} - 2 \sin \frac{α + β}{2} \cos (\frac{α + β}{2} + γ)$
$= 2 \sin \frac{α + β}{2} [2 \sin \frac{α + γ}{2} \sin \frac{γ + β}{2}]$
$4 \sin \frac{α + β}{2} \sin \frac{α + γ}{2} \sin \frac{γ + β}{2} > 0 ∵ 0 < α, β, γ < \frac{π}{2}$
$\Rightarrow \sin α + \sin β + \sin γ > \sin (α + β + γ)$

$\Rightarrow \frac{\sin (α + β + γ)}{\sin α + \sin β + \sin γ} < 1$