Question

How will you prove the formula `cos(A-B)=cosAcosB+sinAsinB` using formula of vector product of two vectors?

Answer 1

As below

Explanation:

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`