Mathematics

Problem

59. The hyperbolic sine function is defined by

\sinh (x)=\left(e^{x}-e^{-x}\right) / 2

. Prove that sinh is an odd function. 60. The hyperbolic cosine function is defined by

\cosh (x)= \left(e^{x}+e^{-x}\right) / 2

. Prove that

\cosh

is an even function. 61. The hyperbolic tangent function is defined by

\tanh (x)= \left(e^{x}-e^{-x}\right) /\left(e^{x}+e^{-x}\right)

. (a) Prove that

\tanh (x)=\sinh (x) / \cosh (x)

. (b) Prove that tanh is an odd function. (c) Prove that

f(x)=1+\tanh (x)

is a logistic function.