Question

How will you prove that cos(A+B)=cosAcosB-sinAsinB using cross product of two vectors?

Answer 1

Let there be two vectors a and b.
Also, a inclined with x-axis with angle A and b inclined with y-axis with angle B.
a = cos A i + sin A j and
b = cos (90 − B) i + sin (90 − B) j = sin B i + cos B j

Let's find cross product of two vectors:
a x b = (cos A i + sin A j) x (sin B i + cos B j)
|a||b|sinθ k = cosAcosB(i × j) + sinAsinB(j × i)

As a and b are unit vectors. Also  θ = {90 - (A+B)} 
sinθ k = cosAcosB k − sinAsinB k
Sin {90 - (A+B)} = cosAcosB − sinAsinB 

Thus,
Cos (A + B) = cosAcosB − sinAsinB