Definitions (a) cosh(x) = (e^{x}+e^{x})/2, (b) sinh(x) = (e^{x}e^{x})/2, (c) tanh(x) = sinh(x)/cosh(x) (= (e^{x}e^{x})/(e^{x}+e^{x})). From the definitions, we can easily deduce


(1) cosh^{2}(x)  sinh^{2}(x) = 1,
(2) sinh(2x) = 2cosh(x)sinh(x), Using (1) and (2),(3), we get
(4) sinh(2x) = 2tanh(x)/(1tanh^{2}(x)), Again from the definition, after some calculation, (7) tanh(x+y) = (tanh(x)+tanh(y))/(1+tanh(x)tanh(y)).


Note that, by (c), tanh(x) = (1e^{2x})/(1+e^{2x}) = 1  2/(e^{2x}+1), so that tanh(0) = 0. Also, for x > 0, e^{2x} > 1, and increases with x, so that tanh(x) increases on [0,∞), and tanh(x) tends to 1 as x tends to ∞. Thus tanh(x) is increasing, and maps [0,∞) to [0,1).
It follows that the inverse function arctanh(x) is an

