sin(A-B)=sinAcosB-cosAsinB

Prove that:

sin(A-B)=sinAcosB-cosAsinB

sin\left( \alpha +\beta \right) =sin\alpha cos\beta +cos\alpha sin\beta \\ \\ But:\quad \\ \\ sin(-\theta )=-sin\theta \\ cos(-\theta )=cos\theta \\ tan(-\theta )=-tan\theta \\ \\ Say:\quad \beta =-\phi \\ \\ \therefore \quad sin\left( \alpha +\left( -\phi \right) \right) =sin\alpha cos\left( -\phi \right) +cos\alpha sin\left( -\phi \right) \\ \\ =sin\alpha cos\phi -cos\alpha sin\phi =sin\left( \alpha -\phi \right) \\ \\ If\quad \alpha =A\quad and\quad \phi =B\\ \\ sin\left( A-B \right) =sinAcosB-cosAsinB.

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