Proof of the analogue to the cosine formula

Starting as before with equation (1),
 
cos a = cos b cos c + sin b sin c cos A.........(1)
we have
sin b sin c cos A = cos a − cos b cos c.
 
Using equation (2),
cos c = cos a cos b + sin a sin b cos C........(2)
we obtain
 
sin b sin c cos A = cos a − cos b(cos a cos b + sin a sin b cosC)
 
so that
 
sin b sin c cos A = cos a sin² b − sin a sin b cos b cosC.
 
Dividing throughout by sin b, we get the required formula, namely
 
sin c cos A = cos a sin b − sin a cos b cosC.