The value of $\tan 90^\circ$ is undefined.

In trigonometry, the tangent of an angle in a right triangle is defined as the ratio of the length of the opposite side to that of the adjacent side. This relationship is mathematically expressed as:

For most angles, this ratio yields a specific numerical value. However, at $90^\circ$, the situation becomes more complex.

At $90^\circ$, the angle creates a scenario where the length of the opposite side is infinitely long compared to the length of the adjacent side, which approaches zero. This results in an attempt to divide by zero, a mathematical operation that is undefined. Consequently, $\tan 90^\circ$ does not possess a finite value and is classified as undefined.

When examining the graph of the tangent function, you will observe that as the angle approaches $90^\circ$ from either direction, the value of the tangent function increases or decreases without limit, tending towards positive or negative infinity. This behavior illustrates why we consider $\tan 90^\circ$ as undefined.

Grasping this concept is essential for analyzing the behavior of trigonometric functions and their applications in various mathematical contexts.