Proof:

Let's start by substituting the given values of x, y, and z into the equation:

x = r sin α cos β
y = r sin α sin β
z = r cos α

Step 1: Calculate x^2, y^2, and z^2 individually.

x^2 = (r sin α cos β)^2 = r^2 sin^2 α cos^2 β
y^2 = (r sin α sin β)^2 = r^2 sin^2 α sin^2 β
z^2 = (r cos α)^2 = r^2 cos^2 α

Step 2: Multiply x^2, y^2, and z^2 together.

x^2 y^2 z^2 = (r^2 sin^2 α cos^2 β) (r^2 sin^2 α sin^2 β) (r^2 cos^2 α)

Step 3: Simplify the expression.

x^2 y^2 z^2 = r^2 sin^2 α cos^2 β sin^2 α sin^2 β cos^2 α
= r^2 sin^4 α cos^2 α sin^2 β cos^2 β

Step 4: Apply the trigonometric identity sin^2 θ + cos^2 θ = 1.

x^2 y^2 z^2 = r^2 (1 - cos^2 α) cos^2 α (1 - cos^2 β) cos^2 β
= r^2 cos^2 α (1 - cos^2 α) cos^2 β (1 - cos^2 β)
= r^2 cos^2 α sin^2 α cos^2 β sin^2 β

Step 5: Apply the trigonometric identity sin^2 θ = 1 - cos^2 θ.

x^2 y^2 z^2 = r^2 (1 - cos^2 α) cos^2 α (1 - cos^2 β) cos^2 β
= r^2 sin^2 α sin^2 β cos^2 α cos^2 β

Step 6: Rearrange the terms.

x^2 y^2 z^2 = r^2 (sin α sin β cos α cos β)^2

Step 7: Apply the trigonometric identity sin θ cos θ = 1/2 sin 2θ.

x^2 y^2 z^2 = r^2 (1/2 sin 2α sin 2β)^2
= r^2 (1/4 sin^2 2α sin^2 2β)

Step 8: Apply the trigonometric identity sin^2 θ = (1 - cos 2θ)/2.

x^2 y^2 z^2 = r^2 (1/4 (1 - cos 4α)/2 (1 - cos 4β)/2)
= r^2 (1/4 (1 - cos 4α)(1 - cos 4β)/4)
= r^2 (