First, find \(\frac{dx}{dt}\) and \(\frac{dy}{dt}\) using the product rule:

\(\frac{dx}{dt} = e^{-t} \sin t - e^{-t} \cos t\)

\(\frac{dy}{dt} = e^{-t} \cos t - e^{-t} \sin t\)

Now, find \(\frac{dy}{dx}\) using \(\frac{dy}{dx} = \frac{dy/dt}{dx/dt}\):

\(\frac{dy}{dx} = \frac{e^{-t} \cos t - e^{-t} \sin t}{e^{-t} \sin t - e^{-t} \cos t}\)

Simplify the expression:

\(\frac{dy}{dx} = \frac{\cos t - \sin t}{\sin t - \cos t}\)

\(\frac{dy}{dx} = \frac{\sin t - \cos t}{\cos t - \sin t}\)

Express \(\frac{dy}{dx}\) in terms of \(\tan t\):

\(\frac{dy}{dx} = \tan \left( t - \frac{1}{4} \pi \right)\)