Please help with this math!??

Question

Use the following functions:

cosh(x)=(1/2)(e^x + e^-x)

sinh(x)=(1/2)(e^x - e^-x)

a. Show that, for any x, [cosh(x)]^2 - [sinh(x)]^2 = 1
b. Show that, for any x,
i) sinh(2x)=2 cosh(x) sinh(x), and
ii) cosh(2x)= [cosh(x)]^2 + [sinh(x)]^2.

Please help and show all work??!!!

Answer 1

cosh(x) = (1/2)(e^x + e^-x)

sinh(x) = (1/2)(e^x - e^-x)

a. Show that, for any x, [cosh(x)]^2 - [sinh(x)]^2 = 1


[cosh(x)]^2 - [sinh(x)]^2 =
((1/2)(e^x + e^-x))^2 - ((1/2)(e^x - e^-x))^2 =
(1/4)(e^2x + 2e^xe^-x + e^-2x) - (1/4)(e^2x - 2e^xe^-x + e^-x) =
(1/4)(e^2x + 2e^xe^-x + e^-2x - e^2x + 2e^xe^-x - e^-2x) =
(1/4)(2e^xe^-x + 2e^xe^-x) =
(1/4)4e^(x - x) =
(1/4)4e^0 =
1*1 =
1

The other two are just as straightforward if you expand the right-hand side. Getting to the right-hand side from the left-hand side can be a bit of a challenge.