Prove that ` sin (A+B)= sin A cos B + cos A sin B. `

Explanation

Let OP and OQ be unit vectors making angles A and B with the X-axis such that
`anglePOQ = A +B`
`vec(OP)= hat i cos A + hat j sin A`
`vecOQ=haticosB-hatj sin B`
`now, vecOPxxvecOQ`
`= (1) (1) sin (A+B) (-hatk)`
`= - sin ( A+B) hatk`
`also vec(O)PxxvecOQ= |{:(hati,hatj , hatk),(cosA ,sinA,0),(cosB,-sinB,0):}|`
`(-cos A sinB-sinA cosB)hatk`
`=-(sinA cos B + cosA sinB) hatk`
form (i) and (ii), we get
`sin ( A+B) = sin A cos B + cos A sin B`