I have been working on this for a week with no luck. Thanks in advance for the help!
Follow the hint and write the general point on the curve as $\displaystyle (\cos\theta,\sin\theta,h(\cos\theta,\sin\theta))$, where $\displaystyle h(\cos\theta,\sin\theta) = 4\cos^3\theta\sin\theta - 4\sin^3\theta\cos\theta+1$. Using a bit of trigonometry, you can simplify that to $\displaystyle h(\cos\theta,\sin\theta) = \sin(4\theta)+1$, so that the point on the curve is $\displaystyle (\cos\theta,\sin\theta,\sin(4\theta)+1)$. Differentiate this (with respect to θ) to find the direction of the tangent at that point.