Question

Prove cos(a+b)=cosacosb-sinasinb using vectors

Answer 1

Let 
ˆAandˆB be two unit vectors in the x-y plane such that 
ˆA makes an angle −A and ˆB makes an angle B with x-axis so that the angle between the two is (A+B) 
The unit vectors can be written in Cartesian form as

ˆA=cosAˆi−sinAˆj and ˆB=cosBˆi+sinBˆj....(1)

To prove
cos(A+B)=cosAcosB−sinAsinB

We know that dot product of two vectors is

→A⋅→B=∣∣∣→A∣∣∣∣∣∣→B∣∣∣cosθ

Inserting our unit vectors in the

above; ∣∣∣→A∣∣∣=∣∣∣→B∣∣∣=1 and value of θ=(A+B), we obtain

ˆA⋅ˆB=cos(A+B)

Using equation (1) 


LHS =(cosAˆi−sinAˆj)⋅(cosBˆi+sinBˆj)


From property of dot product we know that only terms containing ˆi⋅ˆiandˆj⋅ˆj are=1 and rest vanish.

∴ LHS=cosAcosB−sinAsinB

Equating LHS with RHS we obtain

cos(A+B)=cosAcosB−sinAsinB