Answer
$$\frac{4}{5 \sqrt{5}}$$
Work Step by Step
We find:
$\frac{\left|\frac{d^{2} y}{d x^{2}}\right|}{\left(\left(\frac{d y}{d x}\right)^{2}+1\right)^{3 / 2}}=\kappa$
$\kappa(x)=\frac{2 \sec ^{2} x|\tan x|}{\left(\sec ^{4} x+1\right)^{3 / 2}}$
$\kappa(\pi / 4)=\frac{4}{5 \sqrt{5}}$
