Among other complexities in math, there are the hyperbolic functions.
The hyperbolic functions are defined in terms of ex and e-x.

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

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

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


coth(x) = 1/ tanh(x)


csch(x) = 1/ sinh(x)


sech(x) = 1/ cosh(x)



Properties

It is easy to prove that

cosh(-x) = cosh(x)

sinh(-x) = - sinh(x)

tanh(-x) = - tanh(x)

cosh(x) > 0

cosh(x) + sinh(x) = ex

cosh(x) - sinh(x) = e-x > 0

cosh2(x) - sinh2(x) = 1

cosh(x) = + sqrt( 1 + sinh2(x) ) >= 1

1 - tanh2(x) = 1/cosh2(x)

1
cosh(x) = ------------------
sqrt(1 - tanh2(x))


tanh(x)
sinh(x) = -------------------
sqrt(1 - tanh2(x))