Slope of tangent for a parametric curve \[ \frac{dy}{dx}=\frac{\frac{dy}{dt}}{\frac{dx}{dt}} \] Given \[ x=e^t\cos t,\qquad y=e^t\sin t \] Differentiate \[ \frac{dx}{dt}=e^t(\cos t-\sin t),\qquad \frac{dy}{dt}=e^t(\sin t+\cos t) \] \[ \Rightarrow \frac{dy}{dx} =\frac{\sin t+\cos t}{\cos t-\sin t} \] At (t=\frac{\pi}{4}) \[ \sin\frac{\pi}{4}=\cos\frac{\pi}{4} \] \[ \frac{dy}{dx} =\frac{\sin t+\cos t}{\cos t-\sin t} =\frac{\sqrt2}{0} =\infty \] So the tangent is vertical.
Angle with x-axis \[ \theta=\frac{\pi}{2} \]