Reflecting the graph of y = cos(x) is across the x-axis is the same as taking y = -cos(x). Reflecting across the y-axis is the same as taking cos(-x). However, cos(-x) is actually equal to cos(x), not -cos(x), so these two are not equivalent (diagram attached).
I have also attached a graph of these functions. The green line is the reflection over the y-axis, which is the same as the original function. The blue line is the reflection over the x-axis.