Posted by tpc at April 19th, 2008

Came across this curious identity while preparing some trigo notes.
\cos (20^\circ) \cos (40^\circ) \cos (80^\circ) =\frac{1}{8}
According to wolfram, it is called Morrie’s Law by Feynman after his childhood friend who showed it to him. Wow, all I need now is some future nobel laureate with a big mouth (no disrespect intended) , so that some cute little identity might be named after me.

It’s easy to prove Morrie, just multiply \sin x to the numerator and denominator for each of the 3 angles. Use the double angle formula for sine and then cancel away with \cos x = \sin (90^\circ -x)

The general formula, easily proven with induction is the following:
\displaystyle 2^n \prod^{n-1}_{k=0} \cos (2^k x) = \frac{\sin(2^n x)}{\sin x}