Answer
Sphere whose center is at $(0,0,\dfrac{1}{2})$ with radius $\dfrac{1}{2}$
Work Step by Step
Conversion of rectangular to spherical coordinates is as follows:
$x=\rho \sin \phi \cos \theta; y=\rho \sin \phi \sin \theta;z=\rho \cos \phi$
and
$\rho=\sqrt {x^2+y^2+z^2}$;
$\cos \phi =\dfrac{z}{\rho}$; $\cos \theta=\dfrac{x}{\rho \sin \phi}$
Here, $\rho =\cos \phi$
This implies that
$\rho^2=\rho \cos \phi$
or, $x^2+y^2+z^2=z$
or, $x^2+y^2+(z-\dfrac{1}{2})^2=\dfrac{1}{4}$
or, $x^2+y^2+(z-\dfrac{1}{2})^2=(\dfrac{1}{2})^2$
This shows that a sphere whose center is at $(0,0,\dfrac{1}{2})$ with radius $\dfrac{1}{2}$
