Answer: see proof below
Explanation:
Use the Pythagorean Identity: cos²Ф + sin²Ф = 1
Use the Difference Identity: cos(A - B) = cosA · cosB + sinA · sinB
Use the Double Angle Identity: cos 2Ф = 1 - 2sin²Ф
Proof LHS → RHS
LHS: (cosA - cosB)² + (sinA - sinB)²
Expand: cos²A - 2cosA · cosB + cos²B + sin²A - 2sinA · sinB + sin²B
Pythagorean Identity: 2 - 2cosA · cosB - 2sinA · sinB
Factor: 2(1 - (cosA · cosB + sinA · sinB))
Difference Identity: 2(1 - (cos(A - B))
Let Ф = (A-B)/2: 2(1 - cos2Ф)
Double Angle Identity: 2(1 - (1 - 2sin²Ф))
Simplify: 2(1 - 1 + 2sin²Ф)
= 4sin²Ф
Substitute (A-B)/2=Ф: 4sin²(A-B)/2
LHS = RHS: 4sin²(A-B)/2 = 4sin²(A-B)/2
