rel="nofollow" href="#ulink_a8f0a15d-8d07-56b2-bdbe-27165083481e">(2.488) gives rise to
if both sides of Eq. (2.496) were instead divided by cosh2 x, one would have gotten
(2.510)
that degenerates to
after taking Eqs. (2.482) and (2.487) on board; the similarity of Eqs. (2.509) and (2.511) with Eqs. (2.469) and (2.471), respectively, is again striking – except for the minus, instead of plus sign.
2.4.2 Argument Transformation Formulae
Equation (2.472), pertaining to sinh x, may be multiplied by Eq. (2.473), pertaining to cosh y, to get
– in much the same way said equations were combined to produce Eq. (2.497) when y = x; after applying the distributive property of multiplication, Eq. (2.512) becomes
(2.513)
or, after recalling the rule of multiplication of exponential functions conveyed by Eq. (2.15),
Algebraic rearrangement permits transformation of Eq. (2.514) to
(2.515)
which is the same as writing
in view of the functional form of Eq. (2.472). A similar rationale may be applied to the product of sinh y by cosh x to get
(2.517)
again based on Eqs. (2.472) and (2.473) – where the two factors may be pooled together as
Eq. (2.518) may also appear as
After splitting its right‐hand side and factoring out −1 in selected exponents, Eq. (2.519) becomes
(2.520)
so Eq. (2.472) may again be invoked to write
in view of Eq. (2.475), one may redo Eq. (2.521) as
Ordered addition of Eqs. (2.516) and (2.522) gives rise to
(2.523)
which simplifies to
following condensation of terms alike – while ordered subtraction unfolds
(2.525)
with combination of terms alike leading to
Note the similarity of Eqs. (2.524) and (2.526) to Eqs. (2.328) and (2.330), respectively.
