This document contains solutions to an exercise with multiple parts: (a) Simplifies the expression (ex + 1)(e-x - 1) - ex(e-x - 1) to e-x - 1. (b) Simplifies the expression (ex - e-x)(ex + e-x) - (ex - e-x)2 to 2(1 - e-2x). (c) Simplifies the expression e-x(ex - e-x) + e-x(ex + e-x) to 2e-2x. (d) Simplifies the expression (ax + a-x)(

1. Exercice 16
(a)
(ex + 1)(e−x − 1) − ex (e−x − 1)
= (e−x − 1)(ex + 1 − ex ) = e−x − 1
(b)
(ex − e−x )(ex + e−x ) − (ex − e−x )2
= (ex − e−x )[ex + e−x − (ex − e−x )]
= (ex − e−x )(ex + e−x − ex + e−x )
= 2e−x (ex − e−x ) = 2 − 2e−2x = 2(1 − e−2x )

2. Exercice 16 (suite..)
(c)
e−x (ex − e−x ) + e−x (ex + e−x )
e−2x
e −x (ex − e−x + ex + e−x ) 2ex · e−x 2
= −2x
= −2x = −2x = 2e2x
e e e

3. Exercice 16 (suite..)
(d)
(ax + a−x )(ax + a−x ) − (ax − a−x )(ax − a−x )
(ax + a−x )2
a2x + 1 + 1 + a−2x − a2x + 1 + 1 − a−2x
=
(ax + a−x )2
4
= x
(a + a−x )2