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##