Mathematics

The key is that $\tan x$ and $\cot x$ are reciprocals with opposite monotonicity: $\tan x$ increases as you move through its domain (so its derivative $\sec^2 x$ is always positive), while $\cot x$ decreases (making its derivative $-\csc^2 x$ always negative)—a direct consequence of how their slopes point in opposite directions.

The key is that $\tan x$ and $\cot x$ are reciprocals with opposite monotonicity: $\tan x$ increases as you move through its domain (so its derivative $\sec^2 x$ is always positive), while $\cot x$ decreases (making its derivative $-\csc^2 x$ always negative)—a direct consequence of how their slopes point in opposite directions.