Transforming Trigonometric Identies 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  
\[i=\sqrt{-1}\]
.
Example:  
\[cos 2x=cos^2x-sin^2x\]
  becomes
\[cosh2x=(coshx)^2-(i \: sinhx)^2=cosh^2x-(-sinh^2x)=cosh^2x+sinh^2x\]