Transforming Trigonometric Tan Identity From Trigonometric to Hyperbolic Form

We can rewrite trigonometric identifies as hyperbolic identities using the transformation
\[sin x \rightarrow i \: sinh x\]

\[cos x \rightarrow cosh x\]

\[tan x \rightarrow i \: tanh x\]

When we use this transformation on the identity  
\[tan A +tanB+tanC=tanA tanB tanC\]
\[A, \: B, \: C\]
  are the interior angles of a triangle, we get
\[i \: tanhA +i \:tanhB+i \: tanhC=(i \:tanhA) (i \: tanhB i \: tanhC)\]

\[i(tanhA +tanB+tanC)=i^3(tanhA tanhB tanhC)=-i(tanhA tanhB tanhC)\]

\[tanhA +tanB+tanC=-tanhA tanhB tanhC\]

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