直角三角形の対角の関係は ∠ａ = π/2 - ∠b である。
よって、下記のような ＋項 と－項 の数が同じ式は、符号を反転させた逆数でも成立する。
| π/4 = 2arctan(1/2)-arctan(1/8)-arctan(1/57)
= -2arctan(2)+arctan(8)+arctan(57) |
| π/4 = 2arctan(1/2)-arctan(1/9)-arctan(1/32)
= -2arctan(2)+arctan(9)+arctan(32) |
| π/4 = 2arctan(1/2)-arctan(1/12)-arctan(1/17)
= -2arctan(2)+arctan(12)+arctan(17) |
| π/4 = 2arctan(1/2)+arctan(1/5)-2arctan(1/7)-arctan(1/18)
= -2arctan(2)-arctan(5)+2arctan(7)+arctan(18) |
| π/4 = 2arctan(1/2)+arctan(1/6)-2arctan(1/7)-arctan(1/43)
= -2arctan(2)-arctan(6)+2arctan(7)+arctan(43) |
| π/4 = 2arctan(1/5)+3arctan(1/6)-3arctan(1/43)-2arctan(1/57)
= -2arctan(5)-3arctan(6)+3arctan(43)+2arctan(57) |
| π/4 = 2arctan(1/2)-2arctan(1/12)+arctan(1/39)-arctan(1/800)
= -2arctan(2)+2arctan(12)-arctan(39)+arctan(800) |
| π/4 = 2arctan(1/2)-2arctan(1/12)+arctan(1/40)-arctan(1/1641)
= -2arctan(2)+2arctan(12)-arctan(40)+arctan(1641) |
| π/4 = 2arctan(1/2)-2arctan(1/17)-arctan(1/39)+arctan(1/800)
= -2arctan(2)+2arctan(17)+arctan(39)-arctan(800) |
| π/4 = 2arctan(1/2)-2arctan(1/17)-arctan(1/40)+arctan(1/1641)
= -2arctan(2)+2arctan(17)+arctan(40)-arctan(1641) |
| π/4 = 3arctan(1/2)-3arctan(1/3)+2arctan(1/5)-2arctan(1/57)
= -3arctan(2)+3arctan(3)-2arctan(5)+2arctan(57) |
| π/4 = 3arctan(1/2)-arctan(1/5)-3arctan(1/7)+arctan(1/57)
= -3arctan(2)+arctan(5)+3arctan(7)-arctan(57) |
| π/4 = 3arctan(1/3)-arctan(1/8)-arctan(1/19)-arctan(1/343)
= -3arctan(3)+arctan(8)+arctan(19)+arctan(343) |
| π/4 = 3arctan(1/3)-arctan(1/8)-arctan(1/23)-arctan(1/83)
= -3arctan(3)+arctan(8)+arctan(23)+arctan(83) |
| π/4 = 3arctan(1/3)-arctan(1/8)-arctan(1/31)-arctan(1/43)
= -3arctan(3)+arctan(8)+arctan(31)+arctan(43) |
| π/4 = 3arctan(1/3)-arctan(1/9)-arctan(1/18)-arctan(1/73)
= -3arctan(3)+arctan(9)+arctan(18)+arctan(73) |
| π/4 = 3arctan(1/3)-arctan(1/13)-arctan(1/18)-arctan(1/21)
= -3arctan(3)+arctan(13)+arctan(18)+arctan(21) |
| π/4 = 5arctan(1/5)-3arctan(1/18)-2arctan(1/57)
= -5arctan(5)+3arctan(18)+2arctan(57) |