prove-that-tanhx-itanx-

Question Number 43452 by mondodotto@gmail.com last updated on 10/Sep/18

prove that

\tanh x = i \tan x

Answered by alex041103 last updated on 10/Sep/18

\tanh x = \frac{e^{x} - e^{- x}}{e^{x} + e^{- x}} = \frac{e^{i \times i x} - e^{- i \times i x}}{e^{i \times i x} + e^{- i \times i x}} =

= \frac{\cos i x + i \sin i x - \cos i x + i \sin x}{\cos i x + i \sin i x + \cos i x - i \sin x} =

= i \frac{2 \sin i x}{2 \cos i x} = i \tan i x

w h e r e e^{i θ} = \cos i θ + i \sin i θ

\Rightarrow \tanh x = i \tan i x