The hyperbolic trig functions cosh and sinh are defined by

coshx = 1 2 (ex + e-x) sinhx = 1 2 (ex- e-x).

Using the definitions sketch the graphs of coshx and sinhx on one diagram and prove the hyperbolic trig identities

cosh2 x - sinh2 x =1 sinh2x = 2sinhx coshx sinh(n+1)x = sinhnx coshx + coshnx sinhx.

Prove, by induction or otherwise, that for x > 0 and positive integral values of n

sinhnx ³ nsinhx.