Hyperbolic Functions The hyperbolic trigonometric

Chapter 5, Problem 5.1.1.196

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Hyperbolic Functions

The hyperbolic trigonometric functions are defined as follows:

\(\sinh x=\frac{e^{x}-e^{-x}}{2} \quad \cosh x=\frac{e^{x}+e^{-x}}{2} \quad \tanh x=\frac{\sinh x}{\cosh x}\)

\(\operatorname{csch} x=\frac{1}{\sinh x} \quad \operatorname{sech} x=\frac{1}{\cosh x} \quad \operatorname{coth} x=\frac{1}{\tanh x}\)

Confirm the identity.

(a) \(\cosh ^{2} x-\sinh ^{2} x=1\)

(b) \(1-\tanh ^{2} x=\operatorname{sech}^{2} x\)

(c) \(\operatorname{coth}^{2} x-1=\operatorname{csch}^{2} x\)