Answer
$$\frac{1}{\sqrt{2}} e^{t}$$
Work Step by Step
We find:
$\frac{\left| y^{\prime \prime}x^{\prime}- x^{\prime \prime}y^{\prime}\right|}{\left(y^{\prime 2}+x^{\prime 2}\right)^{3 / 2}}=\kappa$
$-e^{-t}(\sin t+\cos t)=x^{\prime}(t) \quad (-\sin t+\cos t)e^{-t}=y^{\prime}(t)$
$2 e^{-t}= x^{\prime \prime}(t) ,\sin t \quad y^{\prime \prime}(t)=-2 e^{-t} \cos t$
$x^{\prime} y^{\prime \prime}-y^{\prime} x^{\prime \prime}=2 e^{-2 t}(\sin t+\cos t) \cos t-2 e^{-2 t}(\cos t-\sin t) \sin t=2 e^{-2 t}[1]$
$y^{\prime 2}+x^{\prime 2}=\left[-e^{-t}(\cos t+\sin t)\right]^{2}+\left[e^{-t}(\cos t-\sin t)\right]^{2}=e^{-2 t}[1+1]$
$\kappa=\frac{\left|2 e^{-2 t}\right|}{\left(2 e^{-2 t}\right)^{3 / 2}}=\frac{2 e^{-2 t}}{2 e^{-3 t} \sqrt{2}}=\frac{1}{\sqrt{2}} e^{t}$
