MHB How can the equation be modified to create a perfect ellipse on the graph?

[highmath]

The graph of the equation:
cos(a) + cos(b) = cos(a + b)
show when the equation is valid.

The graph show an ellipses-like and not like a "perfect" ellipse. Why?
If I want to change the equation, what can I do to get a perfect ellipse?!

 

[Evgeny.Makarov]

This equation is discussed on Math.StackExchange. The accepted answer says that it is indeed not an ellipse, but the solutions to any equation $f(x,y)=\text{const}$ around $(x_0,y_0)$ looks like an ellipse for a smooth function $f$ if $\frac{\partial f}{\partial x}(x_0,y_0)=\frac{\partial f}{\partial y}(x_0,y_0)=0$ and $$\begin{vmatrix}\frac{\partial^2f}{\partial x^2}f(x_0,y_0)&\frac{\partial^2f}{\partial x\partial y}f(x_0,y_0)\\\frac{\partial^2f}{\partial x\partial y}f(x_0,y_0)&\frac{\partial^2f}{\partial y^2}f(x_0,y_0)\end{vmatrix}>0$$ (i.e., if $(x_0,y_0)$ is an extremum point of $f$).

 

[Opalg]

If you move the slider for this Desmos graph, you will see that the graph looks very like an ellipse until the parameter $a$ is quite close to $1$.

[DESMOS]advanced: {"version":5,"graph":{"squareAxes":false,"viewport":{"xmin":-1.3747197653766552,"ymin":-0.9204796366940666,"xmax":6.9623451644014835,"ymax":7.416585293084072}},"expressions":{"list":[{"type":"expression","id":"graph1","color":"#2d70b3","latex":"\\cos x\\ +\\ \\cos y\\ -\\ \\cos\\left(x+y\\right)\\ =\\ a","style":"SOLID"},{"type":"expression","id":"2","color":"#388c46","latex":"a=0","hidden":true,"sliderHardMin":true,"sliderHardMax":true,"sliderMin":"-3","sliderMax":"1","sliderInterval":"0.25","style":"SOLID"}]}}[/DESMOS]