Question

show that cos(x+y)=cosx cosy -sinx siny

Answer 1

Heya user,

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

$\limits^{-\ \textgreater \ }_A . \limits^{-\ \textgreater \ }_B = |\limits^{-\ \textgreater \ }_A|| \limits^{-\ \textgreater \ }_B|$cos θ

Inserting our unit vectors in the above; $| ^{-\ \textgreater \ }_A | =| ^{-\ \textgreater \ }_B| = 1$ 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

You can use x, y as substitutes for A & B.......