tanarctan

Tanarctan is a shorthand for the composition of the tangent and arctangent functions. In mathematics, the two common orders of composition are tan(arctan(x)) and arctan(tan(x)), each with distinct, useful properties arising from the way arctan and tan are defined as inverse functions on restricted domains.

The expression tan(arctan(x)) equals x for every real x. This holds because arctan maps the real line onto the interval (-π/2, π/2), where the tangent function is the inverse of arctan. Thus applying tan after arctan recovers the original input.

In contrast, arctan(tan(x)) is not generally equal to x. The tangent function is periodic with period π, so many x values share the same tan(x) value. Arctan, however, returns the principal value in (-π/2, π/2). Consequently, for any x not equal to π/2 + kπ (where tan x is defined), arctan(tan(x)) = x − kπ, chosen so the result lies in (-π/2, π/2). At points x = π/2 + kπ the tangent is undefined, and the composition is itself undefined.

These behaviors reflect the broader principle that composing a function with its inverse depends on domain restrictions. The identity tan(arctan(x)) = x is a straightforward consequence of inverse-function definitions, while arctan(tan(x)) illustrates how periodicity and principal values alter the result.

In practical terms, tanarctan identities are used to simplify expressions, normalize angles, and understand how nested trigonometric functions behave in calculations and software. They emphasise the importance of attention to domains, ranges, and the principal branch when working with inverse trigonometric functions.