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Q.

Find the value of $1 - \sin 10^{\circ} \sin 50^{\circ} \sin 70^{\circ}$ .

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a

\frac{1}{8}

b

\frac{3}{8}

c

\frac{5}{8}

d

\frac{7}{8}

answer is D.

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Detailed Solution

1 - \sin 10^{\circ} \sin 50^{\circ} \sin 70^{\circ} = 1 - \sin 10^{\circ} \cos (90^{\circ} - 50^{\circ}) \cos (90^{\circ} - 70^{\circ})

= 1 - \sin 10^{\circ} \cos 40^{\circ} \cos 20^{\circ} = 1 - \frac{2 \sin 10^{\circ} \cos 10^{\circ} \cos 40^{\circ} \cos 20^{\circ}}{2 \cos 10^{\circ}} = 1 - \frac{\sin 20^{\circ} \cos 40^{\circ} \cos 20^{\circ}}{2 \cos 10^{\circ}}

= 1 - \frac{2 \sin 20^{\circ} \cos 40^{\circ} \cos 20^{\circ}}{2 \times 2 \cos 10^{\circ}} = 1 - \frac{\sin 40^{\circ} \cos 40^{\circ}}{4 \cos 10^{\circ}} = 1 - \frac{2 \sin 40^{\circ} \cos 40^{\circ}}{2 \times 4 \cos 10^{\circ}} = 1 - \frac{\sin 80^{\circ}}{8 \sin 80^{\circ}}

= 1 - \frac{1}{8} = \frac{7}{8}

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