How do I verify a vector identity using Mathematica?

You can use FrenetSerretSystem:

FrenetSerretSystem[{x[s], y[s], z[s]}, s][[-1, -1]] //TeXForm

$\left\{\frac{y'(s) z''(s)-y''(s) z'(s)}{\sqrt{\left(x'(s) y''(s)-x''(s) y'(s)\right)^2+\left(x''(s) z'(s)-x'(s) z''(s)\right)^2+\left(y'(s) z''(s)-y''(s) z'(s)\right)^2}},\frac{x''(s) z'(s)-x'(s) z''(s)}{\sqrt{\left(x'(s) y''(s)-x''(s) y'(s)\right)^2+\left(x''(s) z'(s)-x'(s) z''(s)\right)^2+\left(y'(s) z''(s)-y''(s) z'(s)\right)^2}},\frac{x'(s) y''(s)-x''(s) y'(s)}{\sqrt{\left(x'(s) y''(s)-x''(s) y'(s)\right)^2+\left(x''(s) z'(s)-x'(s) z''(s)\right)^2+\left(y'(s) z''(s)-y''(s) z'(s)\right)^2}}\right\}$

Since N is a built-in symbol, we will use n instead of N. The required formulas are

r = {x[s], y[s], z[s]};
T = D[r, s]
\[Kappa] = Norm[D[T, s]]
n = D[T, s]/\[Kappa]
\[Tau] = Norm[(\[Kappa]*T + D[n, s])]
B = (\[Kappa]*T + D[n, s])/\[Tau]