## Transforming Trigonometric Tan Identity From Trigonometric to Hyperbolic Form

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

$cos x \rightarrow cosh x$

$tan x \rightarrow i \: tanh x$

where
$i=\sqrt{-1}$
.
When we use this transformation on the identity
$tan A +tanB+tanC=tanA tanB tanC$
where
$A, \: B, \: C$
are the interior angles of a triangle, we get
$i \: tanhA +i \:tanhB+i \: tanhC=(i \:tanhA) (i \: tanhB i \: tanhC)$

$i(tanhA +tanB+tanC)=i^3(tanhA tanhB tanhC)=-i(tanhA tanhB tanhC)$

$tanhA +tanB+tanC=-tanhA tanhB tanhC$