Find the Value of (1 + cos π/8)(1 + cos 3π/8)(1 + cos 5π/8)(1 + cos 7π/8)
Question:
\[ (1+\cos \frac{\pi}{8}) (1+\cos \frac{3\pi}{8}) (1+\cos \frac{5\pi}{8}) (1+\cos \frac{7\pi}{8}) \]Solution
Use the identity
\[ 1+\cos\theta = 2\cos^2\frac{\theta}{2} \]Therefore,
\[ P=(1+\cos \frac{\pi}{8}) (1+\cos \frac{3\pi}{8}) (1+\cos \frac{5\pi}{8}) (1+\cos \frac{7\pi}{8}) \] \[ =16 \cos^2\frac{\pi}{16} \cos^2\frac{3\pi}{16} \cos^2\frac{5\pi}{16} \cos^2\frac{7\pi}{16} \] \[ =16 \left( \cos\frac{\pi}{16} \cos\frac{3\pi}{16} \cos\frac{5\pi}{16} \cos\frac{7\pi}{16} \right)^2 \]Using the standard identity
\[ \cos\frac{\pi}{16} \cos\frac{3\pi}{16} \cos\frac{5\pi}{16} \cos\frac{7\pi}{16} = \frac{\sqrt{2}}{8} \]Hence,
\[ P = 16\left(\frac{\sqrt2}{8}\right)^2 \] \[ = 16\cdot\frac{2}{64} \] \[ = \frac12 \]