Memorize how to simplify the hyperbolic sine (sinh) and cosine (cosh) functions first, as these identities form the basis of all the others. Recognize that a single detail sets them apart: sinh(x) = (e^x - e^-x)/2, while cosh(x) = (e^x + e^-x)/2, where "e" is an exponential constant equal in value to approximately 2.718.
Recognize that the hyperbolic tangent (tanh) is equal to the sinh(x)/cosh(x), whereas the hyperbolic cotangent (coth) is equal to its inverse, or cosh(x)/sinh(x). When you divide this out using the simplifications of sinh(x) and cosh(x), you will find: tanh(x) = (e^x - e^-x)/(e^x + e^-x) and coth(x) = (e^x + e^-x)/(e^x - e^-x); further simplification reveals that tanh(x) = (e^2x - 1)/(e^2x +1) and coth(x) = (e^2x +1)/(e^2x - 1).
Take the inverse of the hyperbolic cosine and hyperbolic sine functions to give you the hyperbolic secant (sech) and hyperbolic cosecant (csch) functions. Therefore: sech(x) = 1/cosh(x) = 1/(e^x + e^-x)/2 = 1 x [2/(e^x + e^-x)] = 2/(e^x + e^-x); csch(x) = 1/sinh(x) = 1/(e^x - e^-x)/2 = 1 x [2/(e^x - e^-x)] = 2/(e^x - e^-x).