We note that,
color(blue)((1)cosC-cosD=-2sin((C+D)/2)sin((C-D)/2)
color(violet)((2)sinC-sinD=2cos((C+D)/2)sin((C-D)/2)
Here,
(cosalpha-cosbeta)^2+(sinalpha-sinbeta)^2=4sin^2((alpha-beta)/2)
Using (1)and (2),we get
LHS=(cosalpha-cosbeta)^2+(sinalpha-sinbeta)^2.
=>LHS=[color(blue)(-2sin((alpha+beta)/2)sin((alpha-beta)/2))]^2
color(white)(.................)+[color(violet)(2cos((alpha+beta)/2)sin((alpha-beta)/2))]^2
=>LHS=color(red)(4)sin^2((alpha+beta)/2)color(red)(sin^2((alpha-beta)/2))
color(white)(.................)+color(red)(4)cos^2((alpha+beta)/2)color(red)(sin^2((alpha-beta)/2))
:.LHS=color(red)(4sin^2((alpha-beta)/2)){sin^2(color(green)((alpha+beta)/2))+cos^2(color(green)((alpha+beta)/2))}
But we know that,
sin^2theta +cos^2theta=1,where, theta=color(green)(((alpha+beta)/2))
LHS=4sin^2((alpha-beta)/2){1}
:.LHS=4sin^2((alpha-beta)/2)=RHS