# Inverse Tangent of Imaginary Number

## Theorem

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

## Proof 1

 $\displaystyle y$ $=$ $\displaystyle \tan^{-1} \paren {i x}$ $\displaystyle \implies \ \$ $\displaystyle \tan y$ $=$ $\displaystyle i x$ Definition of Inverse Tangent $\displaystyle \implies \ \$ $\displaystyle i \tan y$ $=$ $\displaystyle - x$ $i^2 = -1$ $\displaystyle \implies \ \$ $\displaystyle \tanh \paren {i y}$ $=$ $\displaystyle -x$ Tangent in terms of Hyperbolic Tangent $\displaystyle \implies \ \$ $\displaystyle i y$ $=$ $\displaystyle \tanh^{-1} \paren {-x}$ Definition of Inverse Hyperbolic Tangent $\displaystyle \implies \ \$ $\displaystyle i y$ $=$ $\displaystyle -\tanh^{-1} x$ Inverse Hyperbolic Tangent is Odd Function $\displaystyle \implies \ \$ $\displaystyle y$ $=$ $\displaystyle i \tanh^{-1} x$ multiplying both sides by $-i$

$\blacksquare$

## Proof 2

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

$\blacksquare$