# Inverse Tangent of Imaginary Number/Proof 2

$\tan^{-1} \left({i x}\right) = i \tanh^{-1} x$
 $\displaystyle \tan^{-1} \left({i x}\right)$ $=$ $\displaystyle \frac i 2 \ln \left({\frac {1 - i \left({i x}\right)} {1 + i \left({ i x}\right)} }\right)$ Arctangent Logarithmic Formulation $\displaystyle$ $=$ $\displaystyle \frac i 2 \ln \left({\frac{1 + x} {1 - x} }\right)$ $\displaystyle$ $=$ $\displaystyle i \tanh^{-1} x$ Definition of Inverse Hyperbolic Tangent
$\blacksquare$