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.