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 Proof: Hyperbolic Trigonometric Identities (Math | Trig | Hyperbolas)

### Hyperbolic Definitions

sinh(x) = ( e x - e -x )/2

csch(x) = 1/sinh(x) = 2/( e x - e -x )

cosh(x) = ( e x + e -x )/2

sech(x) = 1/cosh(x) = 2/( e x + e -x )

tanh(x) = sinh(x)/cosh(x) = ( e x - e -x )/( e x + e -x )

coth(x) = 1/tanh(x) = ( e x + e -x)/( e x - e -x )

cosh 2(x) - sinh 2(x) = 1

tanh 2(x) + sech 2(x) = 1

coth 2(x) - csch 2(x) = 1

### Inverse Hyperbolic Definitions

arcsinh(z) = ln( z + (z 2 + 1) )

arccosh(z) = ln( z (z 2 - 1) )

arctanh(z) = 1/2 ln( (1+z)/(1-z) )

arccsch(z) = ln( (1+(1+z 2) )/z )

arcsech(z) = ln( (1(1-z 2) )/z )

arccoth(z) = 1/2 ln( (z+1)/(z-1) )

### Relations to Trigonometric Functions

sinh(z) = -i sin(iz)

csch(z) = i csc(iz)

cosh(z) = cos(iz)

sech(z) = sec(iz)

tanh(z) = -i tan(iz)

coth(z) = i cot(iz)

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