1. Write the given equations

   $x = 4\sin\left(\frac{\pi}{2}-\omega t\right), \qquad y = 4\sin(\omega t)$

2. Use the trigonometric identity

   $\sin\left(\frac{\pi}{2}-\theta\right)=\cos\theta$

   So,

   $x = 4\cos(\omega t), \qquad y = 4\sin(\omega t)$

3. Eliminate the time parameter

   Square both equations and add:

   $\left(\frac{x}{4}\right)^2 + \left(\frac{y}{4}\right)^2 = \cos^2(\omega t) + \sin^2(\omega t)$

   $\frac{x^2}{16} + \frac{y^2}{16} = 1$

   $x^2 + y^2 = 16$

4. Identify the path

   The equation

   $x^2 + y^2 = 16$

   represents a circle of radius $4$ m centered at the origin.

5. Check options

   • A: circular ✅
   • B: helical ❌ (requires 3D motion)
   • C: parabolic ❌
   • D: elliptical ❌ in the usual classification here, since it is specifically a circle

Final Answer

The path of the particle is circular.