How do you verify #2cos(a+b)sin(a-b) = sin(2a) - sin(2b)#?

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Answer 1 · 2016-07-09T05:01:02

We know the identity

#sin(A+B)=sinAcosB +cosA sinB.....(1)#

Putting #,B = -B # we have

#sin(A-B)=sinAcosB -cosA sinB.....(2)#

Subtracting (2) from (1) we get

#sin(A+B)-sin(A-B)=2cosA sinB#

#=>2cosA sinB=sin(A+B)-sin(A-B)...........(3).#

Now if # A+B = 2a.......(4) #

and #A-B = 2b...........(5)#

Adding (4) and (5)

#2A=2(a+b) => A=(a+b)#

Subtracting (5) from (4) we get

#2B=2(a-b) =>B=(a-b)#

Now inserting the values of A and B in (3) we get

#2cos(a+b)sin(a-b)=sin((a+b)+(a-b))-sin((a+b)-(a-b))#

#2cos(a+b)sin(a-b)=sin(2a)-sin(2b)#

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