Yes, in general rotating a point by converting to polar coordinates, adding the rotation angle and converting back won't give something that's has an obvious exact form.
If you desperately wanted to go from your equation involving tan, you could rewrite as cos(t + pi /3), where t=arctan(0.5).
Tan (pi/3) = sqrt(3), so tan(t+pi/3) = (\sqrt(3)+1/2)/(1-\sqrt{3}/2). Then use cos^2 A = 1/(1+tan^2 A). Wouldn't be a lot of fun, though.