Answer
For $\theta=90^{\circ}$, $\cos \theta=\cos 90^{\circ}=0$
Thus, the function tangent is undefined.
AND
For $\theta=90^{\circ}$, $\sin \theta=\sin 90^{\circ}=1 \neq 0$
Thus, the function cotangent is defined.
Work Step by Step
Since, $\tan \theta=\dfrac{Opposite}{Adjacent}=\dfrac{y}{x}$
and $\tan \theta=\dfrac{\sin \theta }{\cos \theta}$
when $\cos \theta=0$, then the tangent function will be undefined.
For $\theta=90^{\circ}$, $\cos \theta=\cos 90^{\circ}=0$
Thus, the function tangent is undefined.
Also, $\cot \theta=\dfrac{1}{\dfrac{\sin \theta }{\cos \theta}}$
Thus,
$\cot \theta=\dfrac{\cos \theta }{\sin \theta}$
For $\theta=90^{\circ}$, $\sin \theta=\sin 90^{\circ}=1 \neq 0$
Thus, the function cotangent is defined.
