Question

Given the following trigonometric formulae, use Osborn's rule to write down the corresponding hyperbolic function formulae.
${\cos }^2\theta +{\sin }^2\theta =1$

Answer 1

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 ;

$si{n}^2\theta +co{s}^2\theta =1$

$co{s}^2\theta \to cos{h}^2\theta$

$si{n}^2\theta \to -sin{h}^2\theta$

Hence;

Corresponding Hyperbolic identity:

$cos{h}^2\theta -sin{h}^2\theta =1$