How to determine if $e^{-t}(\cos t +i\sin t)$ is periodic

Question

$x(t) =e^{-t} (\cos t+i\sin t)$ determine $x(t)$ is periodic or nonperiodic and the period if its periodic

Answer 1

$|x(t)|=e^{-t}$ is strictly decreasing. Hence not periodic.

Answer 2

It can't be periodic, since the length of the function is strictly decreasing.

Indeed, if $t_1<t_2$, then

$$|x(t_1)| = |e^{-t_1}(\cos(t_1)+i\sin(t_1))| = |e^{-t_1}| > |e^{-t_2}| = |e^{-t_2}(\cos(t_1)+i\sin(t_1))|=|x(t_2)|,$$

where I have used that $(\cos(t)+i\sin(t))$ is always a point on the unit circle, and hence has length $1$.