Inverse Hyperbolic Cotangent of Imaginary Number

Theorem

$\map {\coth^{-1} } {i x} = i \cot^{-1} x$

Proof

 $\displaystyle y$ $=$ $\displaystyle \map {\coth^{-1} } {i x}$ $\displaystyle \leadsto \ \$ $\displaystyle \coth y$ $=$ $\displaystyle i x$ Definition of Inverse Hyperbolic Cotangent $\displaystyle \leadsto \ \$ $\displaystyle i \coth y$ $=$ $\displaystyle - x$ $i^2 = -1$ $\displaystyle \leadsto \ \$ $\displaystyle \map \cot {i y}$ $=$ $\displaystyle x$ Hyperbolic Cotangent in terms of Cotangent $\displaystyle \leadsto \ \$ $\displaystyle i y$ $=$ $\displaystyle \cot^{-1} x$ Definition of Inverse Cotangent $\displaystyle \leadsto \ \$ $\displaystyle y$ $=$ $\displaystyle -i \cot^{-1} x$ multiplying both sides by $-i$

$\blacksquare$