Answer
$${\bf{T}}\left( t \right) = \left\langle { - \cos t,\sin t} \right\rangle {\text{ and }}\kappa \left( t \right) = \frac{1}{{3\left| {\cos t\sin t} \right|}}$$
Work Step by Step
$$\eqalign{
& {\bf{r}}\left( t \right) = \left\langle {{{\cos }^3}t,{{\sin }^3}t} \right\rangle \cr
& {\text{Calculate }}{\bf{v}}\left( t \right) \cr
& {\bf{v}}\left( t \right) = {{\bf{r}}^\prime }\left( t \right) \cr
& {\bf{v}}\left( t \right) = \frac{d}{{dt}}\left\langle {{{\cos }^3}t,{{\sin }^3}t} \right\rangle \cr
& {\bf{v}}\left( t \right) = \left\langle { - 3{{\cos }^2}t\sin t,3{{\sin }^2}t\cos t} \right\rangle \cr
& {\text{Find the unit tangent vector }}{\bf{T}} \cr
& {\bf{T}}\left( t \right) = \frac{{{\bf{v}}\left( t \right)}}{{\left| {{\bf{v}}\left( t \right)} \right|}} \cr
& {\bf{T}}\left( t \right) = \frac{{\left\langle { - 3{{\cos }^2}t\sin t,3{{\sin }^2}t\cos t} \right\rangle }}{{\left| {\left\langle { - 3{{\cos }^2}t\sin t,3{{\sin }^2}t\cos t} \right\rangle } \right|}} = \frac{{\left\langle { - 3{{\cos }^2}t\sin t,3{{\sin }^2}t\cos t} \right\rangle }}{{\sqrt {9{{\cos }^4}t{{\sin }^2}t + 9{{\sin }^4}t{{\cos }^2}t} }} \cr
& {\bf{T}}\left( t \right) = \frac{{\left\langle { - 3{{\cos }^2}t\sin t,3{{\sin }^2}t\cos t} \right\rangle }}{{\sqrt {9{{\cos }^2}t{{\sin }^2}t\left( {{{\cos }^2}t + {{\sin }^2}t} \right)} }} \cr
& {\bf{T}}\left( t \right) = \frac{{\left\langle { - 3{{\cos }^2}t\sin t,3{{\sin }^2}t\cos t} \right\rangle }}{{3\left| {\cos t\sin t} \right|}} \cr
& {\bf{T}}\left( t \right) = \left\langle { - \cos t,\sin t} \right\rangle \cr
& {\text{Therefore}}{\text{, the curvature is}} \cr
& \kappa \left( t \right) = \frac{1}{{\left| {\bf{v}} \right|}}\left| {\frac{{d{\bf{T}}}}{{dt}}} \right| \cr
& \kappa \left( t \right) = \frac{1}{{3\left| {\cos t\sin t} \right|}}\left| {\frac{d}{{dt}}\left\langle { - \cos t,\sin t} \right\rangle } \right| \cr
& \kappa \left( t \right) = \frac{1}{{3\left| {\cos t\sin t} \right|}}\sqrt {{{\cos }^2}t + {{\sin }^2}t} \cr
& \kappa \left( t \right) = \frac{1}{{3\left| {\cos t\sin t} \right|}} \cr
& \cr
& {\text{Summary:}} \cr
& {\bf{T}}\left( t \right) = \left\langle { - \cos t,\sin t} \right\rangle {\text{ and }}\kappa \left( t \right) = \frac{1}{{3\left| {\cos t\sin t} \right|}} \cr} $$
