Double Angle Formulas for Inverse Tangent

$$2 \tan^{-1} x = \tan^{-1} \left( \frac{2x}{1-x^2} \right) \quad \mathrm{if} \quad |x| \lt 1$$
$$2 \tan^{-1} x = \sin^{-1} \left( \frac{2x}{1+x^2} \right) \quad \mathrm{if} \quad |x| \leq 1$$
$$2 \tan^{-1} x = \cos^{-1} \left( \frac{1-x^2}{1+x^2} \right) \quad \mathrm{if} \quad x \geq 0$$