Evaluate $\cos \frac\pi7\cos \frac{3\pi}7\cos\frac{9\pi}7 $

Question

I need help evaluating the expression

$$\cos \frac\pi7\cos \frac{3\pi}7\cos\frac{9\pi}7 $$

Can someone show the steps he or she used to arrive at the answer?

Answer 1

Note

\begin{align} & \cos \frac\pi7\cos \frac{3\pi}7\cos\frac{9\pi}7\\ = &-\cos \frac{8\pi}7\cos \frac{4\pi}7\cos\frac{2\pi}7\\ =& -\cos \frac{8\pi}7\cos \frac{4\pi}7\sin\frac{4\pi}7\cdot\frac1{2\sin\frac{2\pi}7}\\ =& -\cos \frac{8\pi}7\sin\frac{8\pi}7\cdot\frac1{4\sin\frac{2\pi}7}\\ =& -\frac{\sin\frac{16\pi}7}{8\sin\frac{2\pi}7}=-\frac18 \end{align}