2.1.4—Hyperbolic functions

Use cosh²x−sinh²x=1, tanh x=sinh x/cosh x and the exponential definitions to rewrite an equation in a form that can be factored or solved with logarithms.

Check the domain after squaring or taking square roots, and test candidates in the original equation. For expressions such as a cosh x+b sinh x, exponential substitution turns the problem into a quadratic in e^x.

Setting t=e^x>0 converts cosh x= (t+t^{-1})/2 and sinh x=(t−t^{-1})/2; solve the resulting quadratic and retain only positive t before taking ln t.

Algebraic roots of the transformed equation are not all valid x-values; positivity of e^x and the original domain must be checked.