Since I have more mathematical knowledge at my disposile now than I did in during Calc 3 I thought I would dig around for a bit and try to remember why this specific identity related to the unit circle.
What I found was the Witch of Angensi. Having seen the curve and animation on the wiki page I then decided to try and recreate this model with Desmos.
In the end I was able to create a decent model of the Witch of Angensi and I thought I would share the process.
This was actually harder than I thought it would be so I decided to start with a point on the circle and then fill in the rest.
So Starting over again.
We begin by creating an equation for the point on the circle as it moves from -pi to pi.
If you want to expand your knowledge you can extend this model to include different values for the radius like I did in my example at the beginning of this post.