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MathCalculus

Hyperbolic Identities: Verify Three Exponential-Definition Identities

Verify three hyperbolic identities from exponential definitions, including a sum of squares and an integer-power identity with its domain conditions.

Question

Verify these hyperbolic identities directly from the exponential definitions of cosh⁡x\cosh x and sinh⁡x\sinh x. In (c), nn is any integer.

(a) cosh⁡x−sinh⁡x=e−x\cosh x-\sinh x=e^{-x}.

(b) cosh⁡2x+sinh⁡2x=cosh⁡(2x)\cosh^2 x+\sinh^2 x=\cosh(2x).

(c) (cosh⁡x+sinh⁡x)n=cosh⁡(nx)+sinh⁡(nx)(\cosh x+\sinh x)^n=\cosh(nx)+\sinh(nx).

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Evidence boundary

This page verifies all three parts of Exercise 4 in the University of Melbourne bridging notes. The proofs are derived from the exponential definitions for real x and integer n; no numerical trials are offered as a proof.