If we have a curve $\gamma :(a,b)\rightarrow \mathbb{R}^n $. Then we have the same curve but with an arc length parametrisation $\hat{\gamma} $ such that $\gamma =\hat{\gamma } \circ s .$ (So $\hat{\gamma }$ has unit speed with respect to $s$.)
I'm looking at this proof of the general curvature formula which of course is $$\kappa(t)= \frac{||\gamma_{tt}\ \times \ \gamma_t ||}{||\gamma_t ||^3} .$$
In the proof we find $$\kappa (t)=\hat{\gamma }_{ss}(s(t))=\frac{\gamma _{tt}}{||\gamma_t ||^2}-\gamma_t\frac{\gamma_{tt}\cdot \gamma_t }{||\gamma_t ||^4 }. $$
But all of a sudden it jumps from this to $$\kappa ^2(s(t))=\frac{||\gamma_{tt}||^2||\gamma_{t}||^2-(\gamma_{tt}\cdot \gamma_{t})^2}{||\gamma_{t}||^6}. $$
So my question is...why?