Question

Hey all,
I am currently working on the Worked Example of Session 20: Hyperbolic Trig Functions and I am a little confused. Looking at the solution, I am not understanding how one gets from the second line of the second approach to the third line (in relation to some kind of expansion of the exponentials) to solving for sinh(x+y). Would anyone be able to clarify this step more clearly for me? Thank you a bunch.

Answer 1

starting with
2exp(x+y) - 2exp(-x-y)
they expanded that into 4 terms
exp(x+y) - exp(-x-y) + exp(x+y) - exp(-x-y)
they then added +exp(y-x) - exp(y-x) (which is zero, right?)
and also added -exp(x-y) + exp(x-y) (also zero)
to get the mess:
(exp(x+y) - exp(-x-y) +exp(y-x) -exp(x-y) )
+ ( exp(x+y) - exp(-x-y)- exp(y-x)+ exp(x-y) )

each group of 4 terms factors:
(exp(x) + exp(-x)(exp(y) -exp(-y))
+ ( exp(x)-exp(-x)) (exp(y) + exp(-y) )

after re-introducing the ¼ factor I left out we have
cosh(x) sinh(y) + sinh(x)cosh(y)

The idea of introducing +exp(y-x) - exp(y-x)
and -exp(x-y) + exp(x-y) is a trick people have learned to use. It
is not totally obvious, but it's good to know.

Answer 2

just to be clear:
for any value A,
A-A is zero, so you can add it to an expression without changing the expressin's value.
In this case we used
e.g. +exp(y-x) - exp(y-x)

Answer 3

Oh, gotcha. Thank you.