Find the Value of cos(2π/15) cos(4π/15) cos(8π/15) cos(16π/15)
Question:
\[ \cos\frac{2\pi}{15} \cos\frac{4\pi}{15} \cos\frac{8\pi}{15} \cos\frac{16\pi}{15} \]Find its value.
Solution
Use the identity
\[ \cos x \cos 2x \cos 4x \cos 8x = \frac{\sin 16x}{16\sin x} \]Let
\[ x=\frac{2\pi}{15} \]Then
\[ \cos\frac{2\pi}{15} \cos\frac{4\pi}{15} \cos\frac{8\pi}{15} \cos\frac{16\pi}{15} = \frac{\sin\left(\frac{32\pi}{15}\right)} {16\sin\left(\frac{2\pi}{15}\right)} \]Now,
\[ \sin\frac{32\pi}{15} = \sin\left(2\pi+\frac{2\pi}{15}\right) = \sin\frac{2\pi}{15} \]Therefore,
\[ \frac{\sin\left(\frac{32\pi}{15}\right)} {16\sin\left(\frac{2\pi}{15}\right)} = \frac{\sin\left(\frac{2\pi}{15}\right)} {16\sin\left(\frac{2\pi}{15}\right)} = \frac{1}{16} \]