Proof:

To prove the equation sin(A/2) * sin(B/2) * sin(C/2) = 1/4 * sin(π - A/4) * sin(π - B/4) * sin(π - C/4), we will use the trigonometric identities and properties of sine function.

Identity 1: sin(π - x) = sin(x)
Identity 2: sin(2θ) = 2sin(θ)cos(θ)
Property 1: sin(θ) = 2sin(θ/2)cos(θ/2)

Proof:

Step 1: Start with the left-hand side (LHS) of the equation:
LHS = sin(A/2) * sin(B/2) * sin(C/2)

Step 2: Apply Property 1 to each term:
LHS = 2sin(A/4)cos(A/4) * 2sin(B/4)cos(B/4) * 2sin(C/4)cos(C/4)

Step 3: Rearrange the terms:
LHS = (2sin(A/4) * 2sin(B/4) * 2sin(C/4)) * (cos(A/4) * cos(B/4) * cos(C/4))

Step 4: Simplify the product of the sines using Identity 2:
LHS = 8sin(A/4)sin(B/4)sin(C/4) * (cos(A/4) * cos(B/4) * cos(C/4))

Step 5: Use Identity 1 to rewrite cos(π - x) as cos(x):
LHS = 8sin(A/4)sin(B/4)sin(C/4) * (cos(π - A/4) * cos(π - B/4) * cos(π - C/4))

Step 6: Apply Property 1 to each term:
LHS = 8sin(A/4)sin(B/4)sin(C/4) * (2cos(A/8)sin(A/8) * 2cos(B/8)sin(B/8) * 2cos(C/8)sin(C/8))

Step 7: Rearrange the terms:
LHS = (8sin(A/4)sin(B/4)sin(C/4) * 2cos(A/8) * 2cos(B/8) * 2cos(C/8)) * (sin(A/8) * sin(B/8) * sin(C/8))

Step 8: Simplify the product of the cosines:
LHS = (8sin(A/4)sin(B/4)sin(C/4) * cos(A/8) * cos(B/8) * cos(C/8)) * (sin(A/8) * sin(B/8) * sin(C/8))

Step 9: Simplify the product of the sines:
LHS = 1/4 * sin(A/8) * sin(B/8) * sin(C/8) * (sin(A/8) * sin(B/8) * sin(C/8))

Step 10: Use Identity 2 to rewrite sin(2