I know that I am supposed to write alpha = lambdaT + muN + v*B then differentiate then use the fact that {T,N,B} is a basis in R^3. I am just unsure how to write alpha as a combination using the given information. Any help would be greatly appreciated! Thank you!
Since the curve is on sphere, if you differentiate $ || \alpha (s) || = 1 $, you find $ T \cdot \alpha $, so $ \alpha(s) = n(s) N(s) + b(s) B(s) $ for some functions $ n(s), b(s) $.
Now, differentiating $ \alpha = n N + b B $, one finds $ \alpha' = T = n' N + n N' + b' B + b B' $. Using the frenet-serret equations to rewrite $N'$ and $B'$, $$ T = (n' - b \tau) N - n \kappa T + (b' + \tau n)B $$ So, dotting with $ T, N, B $ gives $ n = -\frac{1}{\kappa} $ and $ n' - b \tau = b' + \tau n = 0 $. Now dividing $ n' - \tau b = 0 $ by $ \tau $ and differentiating gives: $$ 0=\left(\left(\frac{1}{\tau}\right) n' \right)' - b' \\ =-\left(\left(\frac{1}{\tau}\right) \left(\frac{1}{\kappa} \right)' \right)' + \tau n \\ =-\left(\left(\frac{1}{\tau}\right) \left(\frac{1}{\kappa} \right)' \right)' - \frac{\tau}{\kappa} $$
It's been a while since I thought about this stuff, so I have feeling I went in a few circles. But there is no other answer, so I'll leave it up to clean this one up.