Composition of Tangent and Arctangent Functions

$\tan(\arctan x) = x$ for x in [0, π], while $\arctan(\tan x) = x$ for x in [-π, π], impacting the accuracy of triangulation methods in surveying.

$\tan(\arctan x) = x$ for all x, while $\arctan(\tan x) = x$ only for x in [-π/2, π/2], which is crucial for accurately determining angles in various coordinate systems and navigational calculations.

$\tan(\arctan x) = x$ only for x in [-1, 1], while $\arctan(\tan x) = x$ for all x, affecting the precision of trigonometric calculations in signal processing.

Both $\tan(\arctan x)$ and $\arctan(\tan x)$ always equal x, making them interchangeable in all practical applications involving angle measurements.