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

$$\tan(\arctan x) = x$$ 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.

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

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