Home page for accesible maths Math 101 Chapter 2: Functions of a real variable

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2.25 Hyperbolic identities

The hyperbolic functions satisfy similar, but different, identities to those that are satisfied by the trigonometric functions–beware the ±\pm signs.

Identities satisfied by hyperbolic functions.

(i) cosh2⁡x-sinh2⁡x=1;\cosh^{2}x-\sinh^{2}x=1;

(ii) cosh2⁡x+sinh2⁡x=cosh⁡2⁢x;\cosh^{2}x+\sinh^{2}x=\cosh 2x;

(iii) 2⁢sinh⁡x⁢cosh⁡x=sinh⁡2⁢x;2\sinh x\,\cosh x=\sinh 2x;

(iv) cosh⁡(x+y)=cosh⁡x⁢cosh⁡y+sinh⁡x⁢sinh⁡y.\cosh(x+y)=\cosh x\cosh y+\sinh x\sinh y.

Remark. To prove functional identities, start from one side and work towards the other, using the functional equation of exp\exp. Do NOT start the proof by assuming that the identity is true and attempting to prove that 1=11=1.