Question : What will be the value of $\sin 10^{\circ}- \frac{4 }{ 3} \sin ^3 10^{\circ}?$
Option 1: $\frac{1}{3 \sqrt{3}}$
Option 2: $\frac{1}{6}$
Option 3: $\frac{1}{2 \sqrt{3}}$
Option 4: $\frac{\sqrt{3}}{6}$
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Correct Answer: $\frac{1}{6}$
Solution :
$\sin 10^{\circ}- \frac{4 }{ 3} \sin ^3 10^{\circ}$
$=\frac{1}{3}(3\sin 10^{\circ}- 4 \sin ^3 10^{\circ})$
$=\frac{\sin3(10^{\circ})}{3}$ [$\because \sin 3 \theta = 3\sin \theta - 4 \sin 3 \theta$]
$=\frac{\sin30^{\circ}}{3}$
$=\frac{\frac{1}{2}}{3}$
$=\frac{1}{6}$
Hence, the correct answer is $\frac{1}{6}$.
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