Question

Hello, what method is being used to expand this expression?

Answer 1

they wrote 2a as a+a
where a in this case is e^(x+y)
similarly for -2 e^(-x-y)
they then introduced
e^(y-x) - e^(y-x) (which adds to zero)
and similarly
e^(x-y) - e(x-y)

Answer 2

I guess my real question is where did e^(y-x) - e^(y-x) and e^(x-y) - e^(x-y) come from or were they just added arbitrarily?

Answer 3

I guess they are aiming to use Euler's formula
https://en.wikipedia.org/wiki/List_of_trigonometric_identities#Relation_to_the_complex_exponential_function
to write the expression in terms of sin and cos

Answer 4

Except they're not complex functions. The first line simplifies to sinh (x+y), the hyperbolic sine. The second line is equivalent to the first line.

What's the context? When extra terms are added that cancel each other, the purpose is usually to regroup the terms, in this case to get the sum of two or more groups of terms.