Proving Hyperbolic Identities

Question

Prove that:

cosh⁶ x - sinh⁶ x=4, where sinh 2x=2

Answer 1

If sinh 2x = 2, (e^2x - e^-2x)/2 = 2, or (e^2x - e^-2x) = 4

Squaring, e^4x - 2 + e^-4x = 16, or e^4x + e^-4x = 18

cosh^6x - sinh^2x = (e^6x + 6 e^4x + 15 e^2x + 20 + 15e^-2x + 6e^-4x + e^-6x)/64 - (e^6x - 6 e^4x + 15 e^2x - 20 + 15e^-2x - 6e^-4x + e^-6x)/64 =

(3 e^4x + 10 + 3e^-4x)/16 =

3/16 (e^4x + e^-4x) + 10/16 = substituting from the above

3/16(18) + 10/16 =

54/16 + 10/16 =

64/16 =

4