Let $c$: $I \to \Bbb{R}^3$ be an arc length parametrized curve with curvature $k(t) \ne$ 0 for all $t \in I$. Show that the torsion of $c$ is given by
$\tau(t)$ = $⟨c'(t) \space \times \space c''(t), \space \space c'''(t)⟩ \over {(k(t))^2}$
From the Frenet equations, I know that:
$\tau(t)$ = ⟨$n'(t)$, $b(t)$⟩
$\kappa(t)$ = || $c'(t)$||
$b(t)$ = $c'(t)n(t)$
$n(t)$ = $\frac{c''(t)}{k}$
However, I can't just plug these in to verify that the equation in the question is true. It does work out if I plug in $\kappa(t)$ instead of $\kappa$ into $n(t)$ = $\frac{c''(t)}{k}$ , but I think it works out incorrectly... despite that I got the answer, $\kappa(t)$ is a function, not a constant (like in the Frenet equations).
I can't just simply write the following, right?:
$\tau(t)$ = ⟨$n'(t)$, $b(t)$⟩
= ⟨$\frac{c''(t)}{k(t)}$, $c'(t)n(t)$⟩
How would I go about solving this equation and taking care of the $\kappa$?