How will you prove the formula ##cos(A-B)=cosAcosB+sinAsinB## using formula of vector product of two vectors?
As below
Let us consider two unit vectors in X-Y plane as follows :
##hata->## inclined with positive direction of X-axis at angles A
## hat b->## inclined with positive direction of X-axis at angles 90+B, where ## 90+B>A##
Angle between these two vectors becomes
##theta=90+B-A=90-(A-B)##,
##hata=cosAhati+sinAhatj##
##hatb=cos(90+B)hati+sin(90+B)##
##=-sinBhati+cosBhatj##
Now
## hata xx hatb=(cosAhati+sinAhatj)xx(-sinBhati+cosBhatj)##
##=>|hata||hatb|sinthetahatk=cosAcosB(hatixxhatj)-sinAsinB(hatjxxhati)##
Applying Properties of unit vectos ##hati,hatj,hatk##
##hatixxhatj=hatk ##
##hatjxxhati=-hatk ##
##hatixxhati= “null vector” ##
##hatjxxhatj= “null vector” ##
and
##|hata|=1 and|hatb|=1″ “”As both are unit vector” ##
Also inserting
##theta=90-(A-B)##,
Finally we get
##=>sin(90-(A-B))hatk=cosAcosBhatk+sinAsinBhatk##
##:.cos(A-B)=cosAcosB+sinAsinB##