For ellipse x = a*cosφ, and y = a*sinφ is not possible. I think it should be
y = b*sinφ.
To find the tangent vector, write the equation of the ellipse in the standard form and proceed.
Well, if you consider a circle to be a special case of an ellipse?
If you meant , then you can do the following: [tex]\frac{dy}{dx}= \frac{\frac{dy}{dt}}{\frac{dx}{dt}= \frac{b cos(t)}{-a sin(t)}[tex]. At , [tex] and so the slope does not exist. The tangent line at is the vertical line x= -a. The unit tangent vector is .