Question

Prove that cos(a+b) = cos a × cos b - sin a × sin b​

Answer 1

Answer:

Let  and B^ be two unit vectors in the x and y

planes, respectively such that  makes an angle −A and B^ 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

Â=cosÂi−sinÂj

and

B^=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

Â.B^=cos(A+B)

Using equation (1)

LHS

=(cosÂi−sinÂj)⋅(cosBˆi+sinBˆj)

From property of dot product we know that only terms containing

î.î and j^⋅j^ = 1

and rest vanish.

∴

LHS=cosAcosB−sinAsinB

Equating LHS with RHS we obtain

cos(A+B)=cosAcosB−sinAsinB

Hence, proved.