I'm reading the following section from the book 'Curves and Surfaces' by Do Carmo, but I'm stuck and after having gone over this like 10 times I'm starting to think it must be a misprint. The problem is with the last equation, the one for $f$. All of a sudden there is a square in the denominator, and I just don't see where that's coming from. The numerators on both sides of the $=$ are equal, so I don't see how that can be a square on one denominator and not on the other.
Confirmation that this is indeed a misprint or an explanation of what's going on would be appreciated.
I'm not sure I understand this but it looks like you found a typo. In the last line,assuming $x_{u t}=x_{t u}$ then from the previous line, we have $x_{u t}=x_{t u}=w'$. From a previous line,$x_t\wedge x_u=\beta' \wedge w+u w'\wedge w$, and since $u w'\wedge w$ is orthogonal to $w'$ we should have $(x_t,x_u,x_{u t})=(x_t\wedge x_u,w')=(\beta' \wedge w+u w'\wedge w,w')=(\beta' \wedge w,w')=(\beta',w,w')$ so in the last line the square in the denominator is a typo.