円周率が 16 arctan(1/5) - 4 arctan(1/239) で求まる理由

tan(2θ)
= tan(θ+θ)
= (tanθ + tanθ)/(1 - tanθtanθ)
= (2 tanθ)/(1 - (tanθ)^2)

tan(2 arctan(1/5))
= (2 tan(arctan(1/5)))/(1 - (tan(arctan(1/5)))^2)
= (2 (1/5)))/(1 - (1/5)^2)
= (2・5)/(5^2 - 1)
= 10/(25 - 1)
= 10/24
= 5/12

tan(4 arctan(1/5))
= tan(2 (2 arctan(1/5)))
= (2 tan(2 arctan(1/5)))/(1 - (tan(2 arctan(1/5)))^2)
= (2 (5/12))/(1 - (5/12)^2)
= (2・5・12)/(12^2 - 5^2)
= 120/(144 - 25)
= 120/119

tan(4 arctan(1/5) - arctan(1/239))
= tan(4 arctan(1/5) + (- arctan(1/239)))
= (tan(4 arctan(1/5)) + tan(- arctan(1/239)))
     /(1 - tan(4 arctan(1/5)) tan(- arctan(1/239)))
= (tan(4 arctan(1/5)) - tan(arctan(1/239)))
     /(1 + tan(4 arctan(1/5)) tan(arctan(1/239)))
= (120/119 - 1/239)/(1 + (120/119) (1/239))
= (120・239 - 1・119)/(1・119・239 + 120・1)
= (28680 - 119)/(28441 + 120)
= 28561/28561
= 1
よって、
4 arctan(1/5) - arctan(1/239) = π/4 + πn (n は整数)
arctan(1) = π/4 より、
0 < arctan(1/5) < π/4 , - π/4 < - arctan(1/239) < 0 だから
- π/4 < 4 arctan(1/5) - arctan(1/239) < π なので、
4 arctan(1/5) - arctan(1/239) = π/4

16 arctan(1/5) - 4 arctan(1/239)
= 4 (4 arctan(1/5) - arctan(1/239))
= 4 (π/4)
= π