Question

cos(A-B) =cosA cosB +sinA sinB​

Answer 1

Step-by-step explanation:

We want to prove that cos(A−B)=cosAcosB+sinAsinB using vectors.

Consider vectors OP→ and OQ→ on the unit circle having angles A and B respectively with the positive X axis.

⇒|OP→|=1 and |OQ→|=1, and also,

the angle between OP→ and OQ→ is A−B.

OP→=cosAi^+sinAj^ and OQ→=cosBi^+sinBj^.

OP→⋅OQ→=|OP→||OQ→|cos(A−B)=cos(A−B).

Also, OP→⋅OQ→=(cosAi^+sinAj^)⋅(cosBi^+sinBj^)

=cosAcosB+sinAsinB.

⇒cos(A−B)=cosAcosB+sinAsinB.