A little help understanding math....

[Albert Chan]

Another proof of acos(z) algorithm, and showed why we need |z+1|, |z-1|

XCas> z := cos(u+i*v)
XCas> x, y := re(z), im(z)

cos(u)*cosh(v), -sin(u)*sinh(v)

|z|² = x²+y²
= cos(u)²*cosh(v)² + sin(u)²*sinh(v)²
= cos(u)²*cosh(v)² + (1-cos(u)²)*(cosh(v)²-1)
= cosh(v)² + cos(u)² - 1

|z±1|² = (x±1)² + y² = (x²+y²+1) ± 2x
= (cosh(v)² + cos(u)²) ± 2*cosh(v)*cos(u)
= (cosh(v) ± cos(u))²

cosh(v) ≥ 1, cos(u) ≤ 1, we take square root of both side:

|z±1| = cosh(v) ± cos(u)