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Question

Given the following trigonometric formulae, use Osborn's rule to write down the corresponding hyperbolic function formulae.
cos2θ+sin2θ=1

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Solution

Osborn's rule is a rule for converting a trigonometric identity into corresponding hyperbolic one.The rule states that one replaces every occurence of sine or cosine with the corresponding hyperbolic sine or cosine and whenever one has a product of two sines ,the product of hyperbolic sines must be negated.
Thus ;
sin2θ+cos2θ=1

cos2θcosh2θ

sin2θsinh2θ

Hence;
Corresponding Hyperbolic identity:

cosh2θsinh2θ=1

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