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A line makes angles $α, β, γ$ with positive axes, then the range of $\sin αsin β + \sin βsin γ + \sin γsin α$ is-------- $(0 \leq α, β, γ < 180 °)$

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$(\frac{- 1}{2}, 2)$

$(\frac{1}{2}, 2)$

$(0, 2)$

(-1, 2]

answer is C.

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Detailed Solution

$β = \frac{π}{2} - α$
But $\cos^{2} α + \cos^{2} β + \cos^{2} γ = 1 \Rightarrow γ = 90^{\circ}$
$\begin{array}{l} (\cos α + \cos β + \cos γ)^{2} \\ = (\cos α + \cos β)^{2} \\ = 1 + \sin 2 α \end{array}$

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