Find the Value of cos(π/5) cos(2π/5) cos(4π/5) cos(8π/5)
Question:
\[ \cos\frac{\pi}{5}\cos\frac{2\pi}{5}\cos\frac{4\pi}{5}\cos\frac{8\pi}{5} \]Solution
Using periodicity of cosine,
\[ \cos\frac{8\pi}{5} = \cos\left(2\pi-\frac{2\pi}{5}\right) = \cos\frac{2\pi}{5} \]Also,
\[ \cos\frac{4\pi}{5} = \cos\left(\pi-\frac{\pi}{5}\right) = -\cos\frac{\pi}{5} \]Therefore,
\[ P= \cos\frac{\pi}{5}\cos\frac{2\pi}{5} \left(-\cos\frac{\pi}{5}\right) \cos\frac{2\pi}{5} \] \[ =- \cos^2\frac{\pi}{5} \cos^2\frac{2\pi}{5} \]Now use the standard identity
\[ \cos\frac{\pi}{5}\cos\frac{2\pi}{5} = \frac{1}{4} \]Hence,
\[ P = -\left(\frac{1}{4}\right)^2 = -\frac{1}{16} \]