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

Problem

sin(x+y)+sin(x−y)=2*sin(x)*cos(y)

Solution

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

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

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

sin(x−y)=sin(x)*cos(y)−cos(x)*sin(y)

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

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

4. Combine like terms by noticing that cos(x)*sin(y) and −cos(x)*sin(y) cancel each other out.

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

5. Simplify the remaining terms to reach the right side of the identity.

2*sin(x)*cos(y)

Final Answer

sin(x+y)+sin(x−y)=2*sin(x)*cos(y)

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