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Question

Prove by vector method : cos(AB)=cosA.cosB+sinA.sinB.

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Solution

Let i and j are unit vector of x and y axis respectively. If OP and OQ makes angle A and B with x-axes respectively
Let ¯¯¯¯¯¯¯¯OC and ¯¯¯¯¯¯¯¯¯OD are unit vector with respect to OP and OQ respectively.
Hence coordinate of C(cosA,sinA) and D(cosA,sinA) and
OC=icosA+jsinA
OC=icosB+jsinB
Here (AB) is the angle between OC and OD we have
cos(AB)=OC.OD|OC||OD|
cos(AB)=[(cosA)^i+(sinA)j]
[(cosB)^i+(sinB)j]
=cosAcosB+sinAsinB.
665689_628649_ans_e9095e3eb7e64e719d7e51d07beb1492.png

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