Trigonometric Functions Having No Holes

Questions
1

When a rational function such as y = (x^2)/x is shown on a graph, there is a hole at x = 0. But, for the rational trigonometric function y = (cos^2(x))/(cos(x)) is shown on a graph, there are no holes at pi/2, 3pi/2, and so on. Why does this occur?

Screen Shot 2022-12-11 at 2.14.48 AM
Screen Shot 2022-12-11 at 2.14.48 AM2940×1912 402 KB

Screen Shot 2022-12-11 at 2.16.50 AM
Screen Shot 2022-12-11 at 2.16.50 AM2940×1912 394 KB

Screen Shot 2022-12-11 at 2.17.10 AM
Screen Shot 2022-12-11 at 2.17.10 AM2940×1912 288 KB

Thank you for your help.

2

I think it has something to do with approximating pi. If you use cos^2(pix)/cos(pix), then the undefined points will show up as undefined. Without this change the point’s x-value never reaches exactly pi/2.