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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.

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Osborne's rule states that: "a trigonometry identity can be converted to an analogous identity for hyperbolic functions by expanding, exchanging trigonometric functions with their hyperbolic counterparts, and then flipping the sign of each term involving the product of two hyperbolic sines."

The key point is the last clause, which isn't followed in the above manipulations. Once this is accounted for, the correct identity results.

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