マチンの公式
マチンの公式は三角関数の公式を用いて証明できる。
二倍角公式を2回用いて、
tan  ( 2 arctan  1 5 ) = 2 tan  ( arctan  1 5 ) 1 − tan 2  ( arctan  1 5 ) = 2 ⋅ 1 5 1 − ( 1 5 ) 2 = 5 12 {\displaystyle \tan \left(2\arctan {\frac {1}{5}}\right)={\frac {2\tan(\arctan {\frac {1}{5}})}{1-\tan ^{2}(\arctan {\frac {1}{5}})}}={\frac {2\cdot {\frac {1}{5}}}{1-({\frac {1}{5}})^{2}}}={\frac {5}{12}}}
tan  ( 4 arctan  1 5 ) = tan  ( 2 ⋅ 2 arctan  1 5 ) = 2 tan  ( 2 arctan  1 5 ) 1 − tan 2  ( 2 arctan  1 5 ) = 2 ⋅ 5 12 1 − ( 5 12 ) 2 = 120 119 {\displaystyle \tan \left(4\arctan {\frac {1}{5}}\right)=\tan \left(2\cdot 2\arctan {\frac {1}{5}}\right)={\frac {2\tan(2\arctan {\frac {1}{5}})}{1-\tan ^{2}(2\arctan {\frac {1}{5}})}}={\frac {2\cdot {\frac {5}{12}}}{1-({\frac {5}{12}})^{2}}}={\frac {120}{119}}}