target="_blank" rel="nofollow" href="#ulink_81fde16f-26fe-59e4-9a66-470f4ef76378">Fig. 2.14c. Finally, note the resemblance between the functional form of Eqs. (2.482) and (2.483) with Eqs. (2.299) and (2.304), respectively – as well as between Eqs. (2.487) and (2.488), on the one hand, and Eqs. (2.309) and (2.314), on the other; this contributes to justify the denomination of (hyperbolic) trigonometric functions.
After squaring both sides of Eqs. (2.472) and (2.473), and then performing ordered subtraction of the result, one obtains
(2.492)
– where Newton’s binomial as per Eqs. (2.237) and (2.238) may be invoked to write
(2.493)
or, equivalently,
because ex e−x = ex−x = e0 = 1; after canceling symmetrical terms, Eq. (2.494) becomes
(2.495)
that readily simplifies to
– which reminds of Eq. (2.442) pertaining to circular functions proper (except for the minus sign). If Eqs. (2.472) and (2.473) are instead multiplied by one another, i.e.
one finds that
(2.498)
with the aid of the distributive property – or else
after lumping factors alike and canceling out symmetrical terms; if Eq. (2.499) is rewritten as
(2.500)
then comparison with Eq. (2.472) allows further reformulation to
(2.501)
that is equivalent to
(2.502)
– identical in form to Eq. (2.328), after setting x = y. This similarity further accounts for the extra labeling of trigonometric ascribed to the hyperbolic functions.
If Eqs. (2.472) and (2.473) are instead employed in parametric form, viz.
coupled with
one may square both sides of Eqs. (2.503) and (2.504) – and then proceed to ordered subtraction thereof to get
after factoring a2 out, Eq. (2.505) becomes
(2.506)
while insertion of Eq. (2.496) supports simplification to
Equation (2.507) is but the analytical equation of a hyperbola – thus backing up the hyperbolic designation for the functions under scrutiny.
Once in possession of Eq. (2.496), one may divide both its sides by sinh2 x to get
(2.508)
where insertion of Eqs. (2.482), (2.483), and
