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I want to demonstrate that:
$ \frac{d\hat{B}}{ds} = -\tau \hat{N}(s) $

knowing that:

$ \frac{d\hat{T}}{ds} = \kappa \hat{N}(s) $ and $\tau = - \frac{d\hat{B}}{ds}.\hat{N}(s)$

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    You should note also that $\hat B$ is defined by $$ \hat B = \hat T \times \hat N $$ – 2017-02-20

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Differentiating $B = T \times N$, $$\dfrac{dB}{ds} = \dfrac{dT}{ds} \times N + T \times \dfrac{dN}{ds}$$ But the first term is $0$, thus $$\dfrac{dB}{ds} = T \times \dfrac{dN}{ds} \perp T$$ Differentiating $B \cdot B = 1$, we find $$\dfrac{dB}{ds} \perp B$$ Thus $\dfrac{dB}{ds}$ must be a multiple of $N$, and knowing $\dfrac{dB}{ds} \cdot N$ does the rest.