Identify the function which represents a nonperiodic motion.

${\mathrm{e}}^{{-\mathrm{\omega t}}^{ }}$ is an exponential function. It does not repeat its value at regular intervals, and therefore, it is not periodic.
A function f(x) is periodic if it follows the condition f(x+T)=f(x) where T is the interval after which it repeats itself.
sin⁡ωt or  $\sin (\mathrm{\omega t}+\mathrm{\pi }/4)$ is a periodic function, as any sin or cos value repeats after a period 2π.
$\sin (\mathrm{\omega t}+2\mathrm{\pi })=\sin \mathrm{\omega t}$

 sin⁡ωt+cos⁡ωt is also a periodic function.
 $\begin{array}{l}\left(\sin \mathrm{\omega t}\frac{1}{\sqrt{2}}+\cos \mathrm{\omega t}\frac{1}{\sqrt{2}}\right)\sqrt{2}\\ =\left(\sin \mathrm{\omega tcos}\frac{\mathrm{\pi }}{4}+\cos \mathrm{\omega tsin}\frac{\mathrm{\pi }}{4}\right)\sqrt{2}\end{array}$
$=\sqrt{2}\cdot \sin \left(\mathrm{\omega t}+\frac{\mathrm{\pi }}{4}\right)$
 which is a periodic function.
So, from the given functions, only e^−ωt is not a periodic function, all others are periodic.