Understanding of definitions of hyperbolic functions

Defining Hyperbolic Functions

Hyperbolic functions are defined in terms of the exponential function. The hyperbolic sine (sinh) and cosine (cosh) are defined as follows:
- sinh x = (e^x - e^-x) / 2
- cosh x = (e^x + e^-x) / 2
Tanh, coth, sech, and csch (hyperbolic tangent, cotangent, secant, and cosecant) are derived from sinh and cosh. They are:
- tanh x = sinh x / cosh x
- coth x = cosh x / sinh x
- sech x = 1 / cosh x
- csch x = 1 / sinh x

Properties of Hyperbolic Functions

Hyperbolic Identity: cosh^2 x - sinh^2 x = 1. This identity is a core feature of hyperbolic functions.
Addition formulas for sinh and cosh closely resemble those of sin and cos, but with a few key differences:
- sinh(x ± y) = sinh x cosh y ± cosh x sinh y
- cosh(x ± y) = cosh x cosh y ± sinh x sinh y
The derivatives of the hyperbolic functions can be easily obtained by differentiation:
- 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
Inverse Hyperbolic Functions have unique properties:
- sinh^-1 x = ln(x + √(x^2 + 1))
- cosh^-1 x = ln(x + √(x^2 - 1)), x ≥ 1
- tanh^-1 x = (1/2) ln((1 + x) / (1 - x)), x < 1