The exact value of $\tan 45^\circ$ is $1$.

To comprehend why $\tan 45^\circ$ equals $1$, we should explore some fundamental concepts in trigonometry. The tangent of an angle in a right-angled triangle is defined as the ratio of the length of the opposite side to the length of the adjacent side. For the angle $45^\circ$, consider a right-angled triangle where the two angles other than the right angle are also $45^\circ$. This configuration results in an isosceles right triangle, where the two legs (the opposite and adjacent sides) are of equal length.

Assuming each leg of this triangle measures $1$ unit, we can calculate the tangent of $45^\circ$ as the ratio of the length of the opposite side to the length of the adjacent side. Since both sides are equal, this ratio can be expressed as $\frac{1}{1}$, which simplifies to $1$.

Another approach to understanding this concept involves the unit circle, which is defined as a circle with a radius of $1$ centered at the origin of a coordinate plane. The coordinates of the point where the terminal side of a $45^\circ$ angle intersects the unit circle are $\left(\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2}\right)$. The tangent of the angle is determined by dividing the y-coordinate by the x-coordinate, which gives us:

Therefore, whether we analyze the properties of an isosceles right triangle or utilize the unit circle, we consistently find that $\tan 45^\circ$ is precisely $1$.