The hyperbolic sine and hyperbolic cosine from R to R are defined by sinh x = ex
ID: 3140262 • Letter: T
Question
The hyperbolic sine and hyperbolic cosine from R to R are defined by sinh x = ex - e-x/2 and cosh x = ex + e-x/2. Show that cosh2 x - sinh2 x = 1. (cosh)'x = sinh x and (sinh)'x = cosh x for all x R. The function sinh is injective with the range R so that its inverse sinh-1 is defined on R. The function cosh is not injective, but when it is restricted to [0, infinity) is injective and so it has an inverse, denoted by cosh-1, which is defined on [l, infinity). Using (a) show that sinh(cosh-1 x) = x2 - 1 for all x 1 and cosh(sinh-1 x) = 1 + x2 for all x R. Using (c) and the inverse function theorem, show that (sinh-1)(x) = 1/1 + x2 for all x R and (cosh-1)'(x) = 1/ x2 - 1 for all x > 1. Find the following indefinite integrals: 1/1 + x2 dx and 1/ x2 - 1 dx. Find explicit formulas for sinh-1 and cosh-1 by solving the equation y = sinh x for x in terms of y, etc.Explanation / Answer
a)
cosh^2 x - sinh^2 x = [(e^x+e^-x)/2]^2 -[(e^x-e^-x)/2]^2
= [(e^x+e^-x)^2 -(e^x-e^-x)^2]/4
= [e^2x + e^-2x + 2 - e^2x - e^-2x + 2 ] / 4
= 4/4
= 1
L.H.S = R.H.S
hence proved
b)
cosh'x = [(e^x + e^-x) /2] '
= (e^x - e^-x )/2
= sinh x
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