Properties of Functions S and C

Let S(x) = (3^x - 3^-x)/2 and
C(x) = (3^x + 3^-x)/2

Show the functions S and C possess the given properties.

(a) [C(x)]^2 - [S(x)]^2 = 1

(b) S(-x) = -S(x) and C(-x) = C(x)

S(x) = (3^x - 3^(-x)) / 2
C(x) = (3^x + 3^(-x)) / 2

Show that
(C(x))² - (S(x))² = 1

[(3^x + 3^(-x)) / 2]² - [(3^x - 3^(-x)) / 2]² = 1
(3^x + 3^(-x))² / 4 - (3^x - 3^(-x))² / 4 = 1
[(3^x + 3^(-x))² - (3^x - 3^(-x))²] / 4 = 1
[(3^(2x) + 2(3^x)(3^(-x)) + 3^(-2x)) - (3^(2x) - 2(3^x)(3^(-x)) + 3^(-2x))] / 4 = 1
[3^(2x) + 2 + 3^(-2x)) - 3^(2x) + 2 - 3^(-2x)] / 4 = 1
[2 + 2] / 4 = 1
4/4 = 1
1 = 1

= = = = = = = =
Show that S(-x) = -S(x), and C(-x) = C(x).

Here you're showing that S(x) is an odd function and that C(x) is even.

S(-x) = (3^(-x) - 3^x) / 2

-S(x) = -(3^x - 3^(-x)) / 2
-S(x) = (-(3^x) + 3^(-x)) / 2
-S(x) = (3^(-x) - 3^x) / 2

Therefore S(-x) = -S(x).

C(-x) = (3^(-x) + 3^x) / 2
C(-x) = (3^x + 3^(-x)) / 2

C(x) = (3^x + 3^(-x)) / 2

Therefore C(x) = C(-x).