Question

Proof using Osborne's rule that cosh2A=Cosh^2A+Sinh^2A

Answer 1

The rule states that one replaces every occurrence of sine or cosine with the corresponding hyperbolic sine or cosine, and wherever one has a product of two sines, the product of the hyperbolic sines must be negated.

$$\cos(2A)=\cos(A+A)=\cos A\cos A-\sin A\sin A$$

By the Osborne's rule

$$\cosh(2A)=\cosh(A+A)$$

$$=\cosh A\cosh A+\sinh A\sinh A$$

Then

$$\cosh(2A)=\cosh^2 A+\sinh^2 A$$