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  4. The value of cos (2π /15)⋅ cos (4π /15)⋅ cos (8 π/15)⋅ cos (16π /15) is equal to

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Q. The value of $ \cos \frac{2\pi }{15}\cdot \cos \frac{4\pi }{15}\cdot \cos \frac{8 \pi}{15}\cdot \cos \frac{16\pi }{15} $ is equal to

ManipalManipal 2010

Solution:

$\cos \frac{2 \pi}{15} \cdot \cos \frac{4 \pi}{15} \cos \frac{8 \pi}{15} \cos \frac{16 \pi}{15}$
$=\frac{1}{2 \sin \frac{\pi}{15}} \cdot 2 \sin \frac{2 \pi}{15} \cdot \cos \frac{2 \pi}{15} \cdot \cos \frac{4 \pi}{15} \cdot \cos \frac{8 \pi}{15} \cos \frac{16 \pi}{15}$
$\Rightarrow \frac{1}{2 \sin \frac{2 \pi}{15}} \cdot \sin \frac{4 \pi}{15} \cdot \cos \frac{4 \pi}{15} \cdot \cos \frac{8 \pi}{15} \cdot \cos \frac{16 \pi}{15}$
$\Rightarrow \frac{1}{4 \sin \frac{2 \pi}{15}} \cdot \sin \frac{8 \pi}{15} \cdot \cos \frac{8 \pi}{15} \cdot \cos \frac{16 \pi}{15}$
$=\frac{1}{8 \sin \frac{2 \pi}{15}} \cdot \sin \frac{16 \pi}{15} \cdot \cos \frac{16 \pi}{15}$
$=\frac{1}{16 \sin \frac{2 \pi}{15}} \cdot \sin \frac{32 \pi}{15}$
$=\frac{1}{16 \sin \frac{2 \pi}{15}} \cdot \sin \left(2 \pi+\frac{2 \pi}{15}\right)$
$=\frac{1}{16 \sin \frac{2 \pi}{15}} \cdot \sin \frac{2 \pi}{15}$
$=\frac{1}{16}$