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Draw a right angled triangle OPQ with OP the base, PQ the altitude and OQ the hypotenuse.
Draw a second right angled triangle OQR with OQ the base, QR the altitude and OR the hypotenuse.
Drop a perpendicular from R to intercept OP at T. The line RT crosses OQ at U.
Draw a perpendicular from Q to intercept RT at S.
Let angle POQ be A and angle QOR be B.
Angle OUT = angle QUR therefore angle URQ = A
sin (A + B) = TR/OR = (TS + SR)/OR = (PQ + SR)/OR = (PQ/OQ x OQ/OR) + (SR/QR x QR/OR) = sin A cos B + cos A sin B.
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