1. The problem is to find the value of $\tan 90^\circ$. 2. Recall that the tangent function is defined as $\tan \theta = \frac{\sin \theta}{\cos \theta}$. 3. At $90^\circ$, $\sin 90^\circ = 1$ and $\cos 90^\circ = 0$. 4. Substituting these values, we get $\tan 90^\circ = \frac{1}{0}$. 5. Division by zero is undefined in mathematics, so $\tan 90^\circ$ is undefined. 6. Therefore, the value of $\tan 90^\circ$ does not exist; it is undefined.