Hailey is shopping at a department store during a 20% off everything sale. She also has a coupon for $5.00 off the sale amount. Hailey wants to keep her total under $65.00 before tax, so she creates this inequality: 0.80x − $5.00 ≤ $65.00.

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) = .