数学圆周率计算马青公式π/4=4arctan1/5-arctan1/239如何得出的?

设
x = arctan A
tan x = A
tan 2x = (2 tan x) / (1 - tan^2 x)
tan 2x = (2A) / (1 - A^2)
2x = arctan ((2A) / (1 - A^2))
2 arctan A = arctan ((2A) / (1 - A^2))
4 arctan A = 2 arctan ((2A) / (1 - A^2))
A=1/5
4 arctan (1/5) = 2 arctan ((2/5) / (1 - 1/25))=2 arctan (5/12)
2 arctan (5/12) - arctan (1/239)
= arctan ((2*5/12) / (1 - 25/144)) - arctan (1/239)
= arctan (120/119) - arctan (1/239)
= arctan ((120/119 - 1/239) / (1 + 120/119 * 1/239)
= arctan (1.0042 / 1.0042)
= arctan (1)
= π/4

可以通过欧拉逆函数或者其他公式、方程得到。欧拉逆函数1：arctana+arctanb=arctan[(a+b)/(1-ab)]欧拉逆函数2：arctana-arctanb=arctan[(a-b)/(1+ab)]公式的证明：由tan4x=(4tanx-4tanx^3)/(1-6tanx^2+tanx^4),可算得 tan(4arctan1/5)=120/119又由正切和角公式...