I believe $\cot x$ should be defined as $ \dfrac{\cos x}{\sin x}$ and not as $\dfrac{1}{\tan x}$
Because $\cot x$ and $\dfrac{1}{\tan x}$ aren't even the same function, they have different domains.
So for instance we know $\cot (π/2) = 0$ but $\dfrac{1}{ \tan (π/2)}$ is not even defined.
Am I correct in believing this?