\[sin x \rightarrow i \: sinh x\]
\[cos x \rightarrow cosh x\]
\[tan x \rightarrow i \: tanh x\]
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\[i \: tanhA +i \:tanhB+i \: tanhC=(i \:tanhA) (i \: tanhB i \: tanhC)\]
\[i(tanhA +tanB+tanC)=i^3(tanhA tanhB tanhC)=-i(tanhA tanhB tanhC)\]
\[tanhA +tanB+tanC=-tanhA tanhB tanhC\]