Answer
$\dfrac{1}{(\sqrt 2e^t)}$
Work Step by Step
As we are given that $x=e^t \cos t, y=e^t \sin t $
This yields, $x'=e^t \cos t-e^t \sin t, y'=e^t \sin t+e^t \cos t$
Also, $x''=-2e^t \sin t, y''=2e^t \cos t$
$\kappa=\dfrac{|\dot{x} \ddot{y}-\dot{y} \ddot{x}|}{[\dot{x}^2+\dot{y}^2]^{3/2}}$
Thus, $\kappa=\dfrac{|(e^t \cos t-e^t \sin t)(2e^t \cos t)-(e^t \sin t+e^t \cos t)(-2e^t \sin t)|}{[(e^t \cos t-e^t \sin t)^2+(e^t \sin t+e^t \cos t)^2]^{3/2}}$
or, $\kappa=\dfrac{|2e^t (\cos^2 t+sin^2t)|}{[(2e^t (\cos^2 t+sin^2t)]^{3/2}}$
or, $\kappa=\dfrac{|2e^t (1)|}{[2e^t (1)]^{3/2}}$
Hence, the result.
$\kappa=\dfrac{1}{(\sqrt 2e^t)}$
