Given parametric equations:

$$x = e^t \cos t,\quad y = e^t \sin t$$ To find the angle of the tangent at $t = \frac{\pi}{4}$, compute the slope:
$$\frac{dy}{dx} = \frac{\frac{dy}{dt}}{\frac{dx}{dt}} = \frac{e^t(\sin t + \cos t)}{e^t(\cos t - \sin t)} = \frac{\sin t + \cos t}{\cos t - \sin t}$$ At $t = \frac{\pi}{4}$,
$$\sin\left(\frac{\pi}{4}\right) = \cos\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2}$$ So, $$\frac{dy}{dx} = \frac{\frac{\sqrt{2}}{2} + \frac{\sqrt{2}}{2}}{\frac{\sqrt{2}}{2} - \frac{\sqrt{2}}{2}} = \frac{\sqrt{2}}{0}$$ The slope is undefined, which means the tangent is vertical.

Final Answer: The angle with the x-axis is $$\boxed{90^\circ}$$