# Simple trigonometry identity

It has been more than sixty years since I studied trigonometry, and I cannot come up with a fundamental proof that the product of cos(20), cos(40) and cos(80) is exactly 0.125. (Arguments in degrees, of course.) Can anyone help?

$$\begin{split} \cos(20)\cos(40)\cos(80) &= \frac{\sin(20)\cos(20)\cos(40)\cos(80)}{\sin(20)} \\ &= \frac{0.5\sin(40)\cos(40)\cos(80)}{\sin(20)} \\ &= \frac{0.25\sin(80)\cos(80)}{\sin(20)} \\ &= \frac{0.125\sin(160)}{\sin(20)} \\ &= \frac{0.125\sin(20)}{\sin(20)} \\ &= 0.125 \\ \end{split}$$