The equation given in spherical coordinates is:

$\rho = 1 - \cos(\phi)$

To convert this into rectangular (Cartesian) coordinates, we use the following relationships between spherical and Cartesian coordinates:

1. $\rho^2 = x^2 + y^2 + z^2$ (the radial distance)
2. $\cos(\phi) = \frac{z}{\rho}$ (the polar angle)

Step 1: Substitute $\rho = 1 - \cos(\phi)$ into the expressions for Cartesian coordinates.

From $\cos(\phi) = \frac{z}{\rho}$, we substitute into the given equation:

$\rho = 1 - \frac{z}{\rho}$

Step 2: Multiply both sides by $\rho$ to eliminate the denominator:

$\rho^2 = \rho - z$

Step 3: Replace $\rho^2$ with $x^2 + y^2 + z^2$ since $\rho^2 = x^2 + y^2 + z^2$:

$x^2 + y^2 + z^2 = \sqrt{x^2 + y^2 + z^2} - z$

Step 4: Square both sides to remove the square root:

$(x^2 + y^2 + z^2)^2 = (x^2 + y^2 + z^2) - z^2$

Thus, the equation in rectangular coordinates is now fully expressed.

Would you like further clarification on any step or detail? Here are 5 related questions to explore further:

1. How does the process change when converting other spherical equations to Cartesian?
2. Can this equation represent any specific shape in space?
3. What would the equation look like in cylindrical coordinates?
4. How do the spherical coordinate relations simplify for certain symmetries?
5. What are the physical interpretations of spherical vs. Cartesian coordinates?

Tip: When converting between coordinate systems, always ensure to properly replace all variables based on the relationships between the systems to avoid errors.