I have been asked to convert a circular trigonometric identity into its corresponding hyperbolic version using Osborn’s rule.
The identity is $\cos2x = 1-2\sin^2x$
I know that $\cos2x = \cos^2x-\sin^2x $
Then converting to hyperbolic version gives: $\cosh^2x-\sinh^2x = 1$
Therefore left hand side conversion is $\cos2x = 1$
However I have been told that the conversion for the right hand side is $$1-2\sinh^2x = 1+2\sin^2x$$
It is unclear to me how this is possible, I would appreciate very much if someone could explain the reasoning behind this.