Verify the Identity sin(x+y)-sin(x-y)=2cos(x)sin(y)

Problem

sin(x+y)−sin(x−y)=2*cos(x)*sin(y)

Solution

1. Apply the sum identity for sine to the first term sin(x+y)

sin(x+y)=sin(x)*cos(y)+cos(x)*sin(y)

2. Apply the difference identity for sine to the second term sin(x−y)

sin(x−y)=sin(x)*cos(y)−cos(x)*sin(y)

3. Substitute these expansions back into the left side of the original equation.

sin(x+y)−sin(x−y)=(sin(x)*cos(y)+cos(x)*sin(y))−(sin(x)*cos(y)−cos(x)*sin(y))

4. Distribute the negative sign through the second set of parentheses.

sin(x+y)−sin(x−y)=sin(x)*cos(y)+cos(x)*sin(y)−sin(x)*cos(y)+cos(x)*sin(y)

5. Combine like terms by canceling sin(x)*cos(y) and adding the cos(x)*sin(y) terms.

sin(x+y)−sin(x−y)=2*cos(x)*sin(y)

Final Answer

sin(x+y)−sin(x−y)=2*cos(x)*sin(y)

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