I have a 3D curve parametrized by the arc length, $r(s)$, which I compute the curvature $\kappa$ and de torsion $\tau$ as: \begin{equation} \kappa(s) = ||\frac{dT}{ds}|| = ||r''(s)||\\ \tau (s) = \frac{(r'(s) \times r''(s)) \cdot r'''(s)}{\kappa(s)^2} = \frac{(r'(s) \times r''(s)) \cdot r'''(s)}{||r''(s)||^2} \end{equation}
Using the Lancret theorem I can prove that the curve is a helix, by proving that the relation between curvature and torsion is constant:
\begin{equation} \frac{\kappa}{\tau} = \frac{\frac{ \cos^2(\alpha)}{a}}{ \frac{\cos^3(\alpha)\sin(\alpha)}{a^2}} \\ \frac{\kappa}{\tau} = \frac{a}{\cos \left(\alpha\right)\sin \left(\alpha\right)} \end{equation}
Which is constant for any value of $a$ and $\alpha$, so I prove that the curve is a helix. But I need to demonstrate that "the tangent vector $(T=r'(s))$ of the curve maintains a constant angle between it and a prescribed direction along the curve", which for me is sufficient by saying that the curve is a helix, but I want to demonstrate it mathematically.