Inverse Tangent of Imaginary Number/Proof 2

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Theorem

$\tan^{-1} \left({i x}\right) = i \tanh^{-1} x$


Proof

\(\displaystyle \tan^{-1} \left({i x}\right)\) \(=\) \(\displaystyle \frac i 2 \ln \left({\frac {1 - i \left({i x}\right)} {1 + i \left({ i x}\right)} }\right)\) 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

$\blacksquare$