Editing Tangent half-angle formula (section)

==The Gudermannian function==
{{Main|Gudermannian function}}

Comparing the hyperbolic identities to the circular ones, one notices that they involve the same functions of {{math|''t''}}, just permuted. If we identify the parameter {{math|''t''}} in both cases we arrive at a relationship between the circular functions and the hyperbolic ones. That is, if

<math display="block">t = \tan\tfrac12 \varphi = \tanh\tfrac12 \psi</math>

then

<math display="block">\varphi = 2\arctan \bigl(\tanh \tfrac12 \psi\,\bigr) \equiv \operatorname{gd} \psi.</math>

where {{math|gd(''ψ'')}} is the [[Gudermannian function]]. The Gudermannian function gives a direct relationship between the circular functions and the hyperbolic ones that does not involve complex numbers. The above descriptions of the tangent half-angle formulae (projection the unit circle and standard hyperbola onto the {{math|''y''}}-axis) give a geometric interpretation of this function.