Answer:
see explanation
Step-by-step explanation:
Using the sum/ difference → product formula
cos x - cos y = - 2sin( [tex]\frac{x+y}{2}[/tex])sin ([tex]\frac{x-y}{2}[/tex] )
sin x - sin y = 2cos ([tex]\frac{x+y}{2}[/tex] )sin ([tex]\frac{x-y}{2}[/tex] )
Given
(cosA - cosB)² + (sinA - sinB )²
= [ - 2sin([tex]\frac{A+B}{2}[/tex])sin([tex]\frac{A-B}{2}[/tex] ) ]² + [ 2cos([tex]\frac{A+B}{2}[/tex] )sin([tex]\frac{A-B}{2}[/tex] ) ]²
= 4sin² ([tex]\frac{A+B}{2}[/tex] )sin² ([tex]\frac{A-B}{2}[/tex] ) + 4cos² ([tex]\frac{A+B}{2}[/tex] )sin² ( [tex]\frac{A-B}{2}[/tex] )
= 4sin² ([tex]\frac{A-B}{2}[/tex] )[ sin² ( [tex]\frac{A+B}{2}[/tex] ) + cos² ( [tex]\frac{A+B}{2}[/tex] ) ← sin²x + cos²x = 1
= 4sin² ( [tex]\frac{A-B}{2}[/tex] ) × 1
= 4sin² ( [tex]\frac{A-B}{2}[/tex] ) = right side ⇒ proven
