Prove by vector method : cos(A−B)=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 (A−B) is the angle between →OC and →OD we have cos(A−B)=→OC.→OD|→OC||→OD| ∴cos(A−B)=[(cosA)^i+(sinA)j] [(cosB)^i+(sinB)j] =cosAcosB+sinAsinB.