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Question: \(\sin\pi/14.\sin 3\pi/14.\sin 5\pi/14.\sin 7\pi/14.\sin 9\pi ⥂ /1 ⥂ 4.\sin 11\pi/14.\sin 13\pi/14\)...

sin⁡π/14.sin⁡3π/14.sin⁡5π/14.sin⁡7π/14.sin⁡9π⥂/1⥂4.sin⁡11π/14.sin⁡13π/14\sin\pi/14.\sin 3\pi/14.\sin 5\pi/14.\sin 7\pi/14.\sin 9\pi ⥂ /1 ⥂ 4.\sin 11\pi/14.\sin 13\pi/14is equal

To

A

18\frac{1}{8}

B

116\frac{1}{16}

C

132\frac{1}{32}

D

164\frac{1}{64}

Answer

164\frac{1}{64}

Explanation

Solution

sin⁡π14.sin⁡3π14.sin⁡5π14.sin⁡7π14.sin⁡9π14.sin⁡11π14.sin⁡13π14\sin\frac{\pi}{14}.\sin\frac{3\pi}{14}.\frac{\sin 5\pi}{14}.\sin\frac{7\pi}{14}.\sin\frac{9\pi}{14}.\sin\frac{11\pi}{14}.\sin\frac{13\pi}{14}= sin⁡π14.sin⁡3π14.sin⁡5π14×1×sin⁡(π−5π14).sin⁡(π−3π14).sin⁡(π−π14)\sin\frac{\pi}{14}.\sin\frac{3\pi}{14}.\sin\frac{5\pi}{14} \times 1 \times \sin\left( \pi - \frac{5\pi}{14} \right).\sin\left( \pi - \frac{3\pi}{14} \right).\sin\left( \pi - \frac{\pi}{14} \right) = [sin⁡π14.sin⁡3π14.sin⁡5π14.sin⁡7π14]2=164\left\lbrack \sin\frac{\pi}{14}.\sin\frac{3\pi}{14}.\sin\frac{5\pi}{14}.\sin\frac{7\pi}{14} \right\rbrack^{2} = \frac{1}{64}.