Mathematics

The unit circle is the perfect visual tool here: as you trace around the circle, the $x$-coordinate of each point is exactly $\cos x$. Since the circle has radius 1 and is centered at the origin, the $x$-coordinate can never go further left than $-1$ or further right than $+1$. Moreover, by continuity, the cosine function actually reaches every value in between as you move around the circle. That's why the range is the complete closed interval $[-1,1]$, not just some subset of it.

The unit circle is the perfect visual tool here: as you trace around the circle, the $x$-coordinate of each point is exactly $\cos x$. Since the circle has radius 1 and is centered at the origin, the $x$-coordinate can never go further left than $-1$ or further right than $+1$. Moreover, by continuity, the cosine function actually reaches every value in between as you move around the circle. That's why the range is the complete closed interval $[-1,1]$, not just some subset of it.