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

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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

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

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

Verified

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