Answer
$$
\kappa(x) =\frac{|\cos x|}{\left(1+\sin^2 x\right)^{3 / 2}},
\quad
\kappa(0)= 1.
$$
Work Step by Step
We have $$
\kappa(x)=\frac{\left|f^{\prime \prime}(x)\right|}{\left(1+f^{\prime}(x)^{2}\right)^{3 / 2}}=\frac{|\cos x|}{\left(1+\sin^2 x\right)^{3 / 2}},
$$
and hence
$$
\kappa(0)=\frac{|\cos 0|}{\left(1+\sin^20\right)^{3 / 2}}=1.
$$
