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  4. The value of sin 10° ⋅ sin 30° ⋅ sin 50°- sin 70° is

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Q. The value of $\sin 10^{\circ} \cdot \sin 30^{\circ} \cdot \sin 50^{\circ}-\sin 70^{\circ}$ is

KCETKCET 2011Trigonometric Functions

Solution:

$\sin 10^{\circ} \cdot \sin 30^{\circ} \cdot \sin 50^{\circ} \cdot \sin 70^{\circ}$
$=\frac{1}{2} \cdot \sin 10^{\circ} \cdot \frac{1}{2}\left(2 \sin 70^{\circ} \cdot \sin 50^{\circ}\right)$
$=\frac{1}{2} \sin 10^{\circ} \cdot \frac{1}{2}\left\{\cos \left(70^{\circ}, 50^{\circ}\right)\right.\\ \left.-\cos \left(70^{\circ}+50^{\circ}\right)\right\}$
$=\frac{1}{2} \sin 10^{\circ} \cdot \frac{1}{2}\left\{\cos 20^{\circ}-\cos 120^{\circ}\right\}$
$=\frac{1}{2} \sin 10^{\circ} \cdot \frac{1}{2}\left(\cos 10^{\circ}+\frac{1}{2}\right)$
$=\frac{1}{4} \sin 10^{\circ} \cdot \cos 20^{\circ}+\frac{1}{8} \sin 10^{\circ}$
$=\frac{1}{4} \cdot \frac{1}{2}\left(\sin 30^{\circ}-\sin 10^{\circ}\right)+\frac{1}{8} \cdot \sin 10^{\circ}$
$=\frac{1}{8} \cdot \sin 30^{\circ}-\frac{1}{8} \sin 10^{\circ}+1 / 8 \sin 10^{\circ}$
$=\frac{1}{8} \cdot \frac{1}{2}-0=\frac{1}{16}$