2tan^(-1)(x) = sin^(-1)(2$\frac{x}{1+x^2}$) when |x| <= 1. 2tan^(-1)(x) = cos^(-1)$\frac{(1-x^2}{(1+x^2)}$) when x >= 0. 2*tan^(-1)(x) = tan^(-1)(2$\frac{x}{1-x^2}$) when |x| < 1. The conditions are critical — violating them gives wrong answers because the principal value branch changes.