How do you prove 2 sin x cos x = sin (x + y) − sin (x − y)?

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Answer 1 · 2018-03-05T22:40:56

See below.

Identities:

#color(red)bb(sin(A+B)=sinAcosB+cosAsinB)#

#color(red)bb(sin(A-B)=sinAcosB-cosAsinB)#

#RHS#

#sin(x+y)=sinxcosy+cosxsiny#

#sin(x-y)=sinxcosy-cosxsiny#

#sin(x+y)-sin(x-y)#

#=sinxcosy+cosxsiny-(sinxcosy-cosxsiny)#

#=sinxcosy+cosxsiny-sinxcosy+cosxsiny#

#=cancel(sinxcosy)+cosxsinycancel(-sinxcosy)+cosxsiny#

#=2sinxcosy#

#RHS -= LHS#