The exact value of $\cos 90^\circ$ is $0$.

To comprehend why $\cos 90^\circ$ equals $0$, we must explore the unit circle, a fundamental concept in trigonometry. The unit circle is defined as a circle with a radius of $1$, centered at the origin of a coordinate plane. In this context, angles are measured from the positive x-axis, moving counterclockwise around the circle.

In the unit circle, the cosine of an angle corresponds to the x-coordinate of the point where the terminal side of the angle intersects the circle. For instance, at $0^\circ$, the point of intersection is $(1, 0)$, which gives us $\cos 0^\circ = 1$. As the angle increases, this point moves along the circumference of the circle.

When the angle reaches $90^\circ$, the point is located at the top of the circle, specifically at $(0, 1)$. At this position, the x-coordinate is $0$, which is why $\cos 90^\circ = 0$. This value is essential to remember, as it frequently appears in various trigonometric problems and calculations.

Visualizing the unit circle can greatly aid in recalling these values. For specific angles such as $0^\circ$, $90^\circ$, $180^\circ$, and $270^\circ$, the corresponding cosine values are particularly straightforward: $1$, $0$, $-1$, and $0$, respectively. Grasping these fundamental values will facilitate your understanding and problem-solving skills in more complex trigonometric scenarios encountered in your GCSE Mathematics studies.