モリーの法則
正弦関数の倍角の公式より、
sin  ( 2 α ) = 2 sin  ( α ) cos  ( α ) {\displaystyle \sin(2\alpha )=2\sin(\alpha )\cos(\alpha )}
cos  ( α ) {\displaystyle \cos(\alpha )} について解くと
cos  ( α ) = sin  ( 2 α ) 2 sin  ( α ) {\displaystyle \cos(\alpha )={\frac {\sin(2\alpha )}{2\sin(\alpha )}}}
これより
cos  ( 2 α ) = sin  ( 4 α ) 2 sin  ( 2 α ) cos  ( 4 α ) = sin  ( 8 α ) 2 sin  ( 4 α ) ⋮ cos  ( 2 n − 1 α ) = sin  ( 2 n α ) 2 sin  ( 2 n − 1 α ) {\displaystyle {\begin{aligned}\cos(2\alpha )&={\frac {\sin(4\alpha )}{2\sin(2\alpha )}}\\[6pt]\cos(4\alpha )&={\frac {\sin(8\alpha )}{2\sin(4\alpha )}}\\&{}\,\,\,\vdots \\\cos(2^{n-1}\alpha )&={\frac {\sin(2^{n}\alpha )}{2\sin(2^{n-1}\alpha )}}\end{aligned}}}
これらの等式を全て掛け合わせると、