Here is how you would find the horizontal tangents...

So we set it equal to zero to get

where n is an integer

Now to get the cartesian coordinates plug the values of theta into your parametric equations so when n=0 we get pi/4 so

so ONE of the ordered pairs is

This should get you started you will need to use more values of n to find the rest.

Hint there are 4 horizontal tangents and 2 vertical good luck.