Question

Someone please help me!!!!​

Answer 1

Answer:   see proof below

Step-by-step 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   $\checkmark$