Non Permissible values of cot ⁡ ( x )Why is it that the non-permissible values...

one way you can see this is that
$\cot x=\frac{1}{\tan x}=\frac{1}{\frac{\sin x}{\cos x}}=\frac{\cos x}{\sin x}$
which is 0 when $\cos x=0$, and undefined when $\sin x=0$

The tangent of x is defined as the the sine of x divided by the cosine of x, so
$\tan x=\frac{\sin x}{\cos x}$
If x=0 then product is 0/1=0, which is a real solution.
However for
$\cot x=\frac{\cos x}{\sin x}$
Now, if x=0, then the product is 1/0 which is undefined, meaning that a solution does not exist, which means that sinx cannot be equal to 0 for cotx to exist