derived smooth geometry
A trigonometric identity is (formally) a commutative diagram in the category of cartesian spaces and partial functions whose edges are labelled by rational functions (or sometimes algebraic functions) and trigonometric functions.
Slightly more precisely: each rational function is interpreted as a partial function where is the “natural domain” of (see rational function for more discussion); these are partial analytic functions. The basic trigonometric functions are (total) analytic functions . All of these may be interpreted as partial functions , and generate a class of functions by applying the monoidal category structure on the category of partial functions that is induced by the cartesian product on cartesian spaces. A trigonometric identity is then (formally) an equality of morphisms in the monoidal category thus generated.
Of course, this is complete overkill; category theorists are not oblivious to the fact that this is exactly the kind of description lampooned in Linderholm’s Mathematics Made Difficult. It’s just a formal way of covering bases. So let us add that in practice, a trigonometric identity usually involves functions obtained by substituting trigonometric functions into rational functions, or substituting rational linear (affine) functions into trigonometric functions: the class of functions considered is usually fairly limited in scope. Virtually all trigonometric identities can be seen as arising from suitable exponential function identities on complex numbers such as