lf $α, β, γ$ are the angles made by a line with the coordinate axes in the positive direction, then the range of $sin α sin β + sin β sin γ + sin γ sin α$ is

[- \frac{1}{2}, 1]

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[- 1, 2]

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[- \frac{1}{2}, \infty)

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[- 1, \infty)

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Solution

The correct option is B $[- 1, 2]$
If $α, β$ and $γ$ are angles made by a line with the coordinate axes in the positive direction, then
${cos}^{2} α + {cos}^{2} β + {cos}^{2} γ = 1$
$1 - {sin}^{2} α + 1 - {sin}^{2} β + 1 - {sin}^{2} γ = 1$
${sin}^{2} α + {sin}^{2} β + {sin}^{2} γ = 2$
Now , using ${(\sum sin α)}^{2} = \sum {sin}^{2} α + 2 \sum sin α . sin β$
${(\sum sin α)}^{2} \geq 0$
$\Rightarrow \sum {sin}^{2} α + 2 \sum sin α . sin β \geq 0$
$\Rightarrow 2 + 2 \sum sin α . sin β \geq 0$
$\Rightarrow \sum sin α . sin β \geq - 1$ .....(1)
Next use the equation $\sum {(sin α - sin β)}^{2} = 2 \sum {sin}^{2} α - 2 \sum sin α . sin β$
$\sum {(sin α - sin β)}^{2} \geq 0$
$\Rightarrow 2 \sum {sin}^{2} α - 2 \sum sin α . sin β \geq 0$
$\Rightarrow 4 - 2 \sum sin α . sin β \geq 0$
$\Rightarrow \sum sin α . sin β \leq 2$ .....(2)
From (1) and (2), we get the range of $\sum sin α . sin β$ is $[- 1, 2]$
Hence, option 'B' is correct.

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