Ashley, Tamira, and Suzy bought school supplies. Each person spent between $12
$
12
and $15
$
15
.

Ashley bought 3
3
items.
Tamira bought 4
4
items.
Suzy bought 5
5
items.

The prices of the items are given below. Drag items to each box to show what each person could have bought.

Prove (a) cosh2(x) − sinh2(x) = 1 and (b) 1 − tanh 2(x) = sech 2(x). solution (a) cosh2(x) − sinh2(x) = ex + e−x 2 2 − 2 = e2x + 2 + e−2x 4 − = 4 = . (b) we start with the identity proved in part (a): cosh2(x) − sinh2(x) = 1. if we divide both sides by cosh2(x), we get 1 − sinh2(x) cosh2(x) = 1 or 1 − tanh 2(x) = .