Answer
$$\kappa \left( t \right) = \frac{4}{{{{\left( {16{{\sin }^2}t + {{\cos }^2}t} \right)}^{3/2}}}}$$
Work Step by Step
$$\eqalign{
& {\bf{r}}\left( t \right) = 4\cos t{\bf{i}} + \sin t{\bf{j}} \cr
& {\text{Calculate the derivatives }}{\bf{r}}'\left( t \right){\text{ and }}{\bf{r}}{\text{''}}\left( t \right) \cr
& {\bf{r}}'\left( t \right) = \frac{d}{{dt}}\left[ {4\cos t} \right]{\bf{i}} + \frac{d}{{dt}}\left[ {\sin t} \right]{\bf{j}} \cr
& {\bf{r}}'\left( t \right) = - 4\sin t{\bf{i}} + \cos t{\bf{j}} \cr
& and \cr
& {\bf{r}}''\left( t \right) = \frac{d}{{dt}}\left[ { - 4\sin t} \right]{\bf{i}} + \frac{d}{{dt}}\left[ {\cos t} \right]{\bf{j}} \cr
& {\bf{r}}''\left( t \right) = - 4\cos t{\bf{i}} - \sin t{\bf{j}} \cr
& \cr
& {\text{Calculate the cross product }}{\bf{r}}'\left( t \right) \times {\bf{r}}''\left( t \right) \cr} $$
\[\begin{gathered}
{\mathbf{r}}'\left( t \right) \times {\mathbf{r}}''\left( t \right) = \left| {\begin{array}{*{20}{c}}
{\mathbf{i}}&{\mathbf{j}}&{\mathbf{k}} \\
{ - 4\sin t}&{\cos t}&0 \\
{ - 4\cos t}&{ - \sin t}&0
\end{array}} \right| \hfill \\
{\mathbf{r}}'\left( t \right) \times {\mathbf{r}}''\left( t \right) = \left| {\begin{array}{*{20}{c}}
{\cos t}&0 \\
{ - \sin t}&0
\end{array}} \right|{\mathbf{i}} - \left| {\begin{array}{*{20}{c}}
{ - 4\sin t}&0 \\
{ - 4\cos t}&0
\end{array}} \right|{\mathbf{j}} + \left| {\begin{array}{*{20}{c}}
{ - 4\sin t}&{\cos t} \\
{ - 4\cos t}&{ - \sin t}
\end{array}} \right|{\mathbf{k}} \hfill \\
{\mathbf{r}}'\left( t \right) \times {\mathbf{r}}''\left( t \right) = 0{\mathbf{i}} - 0{\mathbf{j}} + \left( {4{{\sin }^2}t + 4{{\cos }^2}t} \right){\mathbf{k}} \hfill \\
{\mathbf{r}}'\left( t \right) \times {\mathbf{r}}''\left( t \right) = 4{\mathbf{k}} \hfill \\
\end{gathered} \]
$$\eqalign{
& {\text{Use the formula }}\left( 3 \right)\,\,\,\kappa \left( t \right) = \frac{{\left\| {{\bf{r}}'\left( t \right) \times {\bf{r}}''\left( t \right)} \right\|}}{{{{\left\| {{\bf{r}}'\left( t \right)} \right\|}^3}}}.{\text{ Thus}}{\text{,}} \cr
& \kappa \left( t \right) = \frac{{\left\| {4{\bf{k}}} \right\|}}{{{{\left\| { - 4\sin t{\bf{i}} + \cos t{\bf{j}}} \right\|}^3}}} \cr
& \kappa \left( t \right) = \frac{4}{{{{\left( {\sqrt {{{\left( { - 4\sin t} \right)}^2} + {{\left( {\cos t} \right)}^2}} } \right)}^3}}} \cr
& \kappa \left( t \right) = \frac{4}{{{{\left( {\sqrt {16{{\sin }^2}t + {{\cos }^2}t} } \right)}^3}}} \cr
& \kappa \left( t \right) = \frac{4}{{{{\left( {16{{\sin }^2}t + {{\cos }^2}t} \right)}^{3/2}}}} \cr} $$
