\(\cos \frac \pi 9 \cos\frac{2\pi}9 \cos \frac{3\pi }9 \cos \frac {4\pi}9\)
\(=\cos \frac \pi 9 \cos\frac{2\pi}9 (\cos \frac{\pi }3 )\cos \frac {4\pi}9\)
\(= \frac12 (\frac 22 \cos \frac \pi 9\cos \frac{2\pi}9) \cos \frac {4\pi}9\)
\(= \frac 14 \left[\cos (\frac \pi 9 - \frac{2\pi}9) + \cos(\frac \pi 9 + \frac{2\pi}9)\right] \cos \frac{4\pi }9\)
\(= \frac 14 \left[ \frac 2 2 \cos \frac \pi 9 \cos \frac{4\pi}9 + \cos\frac \pi3 \cos \frac {4\pi}9\right]\)
\(= \frac 14 \left[\frac 12\cos(\frac \pi 9- \frac{4\pi}9) + \frac 12 \cos (\frac \pi9 + \frac{4\pi}9) + \frac 12 \cos \frac{4\pi}9 \right]\)
\(= \frac14 \left[\frac 12 \cos \frac \pi 3 + \frac 12 \cos \frac{5\pi}9+ \frac 12 \cos \frac {4\pi}9\right]\)
\(= \frac14\left[ (\frac12 \times \frac 12)+ \frac 12 \cos(\pi - \frac{4\pi}9) + \frac 12\cos\frac{4\pi}9\right]\)
\(= \frac14 \left[\frac14 - \frac 12 cos\frac{4\pi}9 + \frac 12 \cos \frac{4\pi}9\right]\)
\(= \frac 1{16}\)
