Hyperbolic Functions

Hyperbolic functions are defined as follows

• `cosh(theta) = 1/2(e^x + e^(-x))`
• `sinh(theta) = 1/2(e^x - e^(-x))`
• `tanh(theta) = (sinh(theta))/(cosh(theta)) = (e^x - e^(-x))/(e^x + e^(-x))`
• `coth(theta) = (cosh(theta))/(sinh(theta)) = (e^x + e^(-x))/(e^x + e^(-x))`
• `sech(theta) = 1/(cosh(theta)) = 2/(e^x + e^(-x))`
• `csch(theta) = 1/(sinh(theta)) = 2/(e^x - e^(-x))`

Identities of hyperbolic functions

• `cosh^2(x) - sinh^2(x) = 1`
• `cosh^2(x) + sinh^2(x) = cosh(2x)`
• `cosh(2x) = cosh^2(x) - 1`
• `cosh(2x) = sinh^2(x) - 1`
• `sinh(2x) = 2sinh(x)cosh(x)`
• `sinh^(-1)(x) = log(x + sqrt(x^2 + 1))`
• `cosh^(-1)(x) = log(x + sqrt(x^2 - 1))` 

Derivatives of hyperbolic functions

• `d/dx sinh(x) = cosh(x)`
• `d/dx cosh(x) = sinh(x)`
• `d/dx tanh(x) = sech^2(x)`
• `d/dx coth(x) = -csch^2(x)`
• `d/dx sech(x) = -sech(x)tanh(x)`
• `d/dx csch(x) = -csch(x)coth(x)`