Simplify sin(a+b)+sin(a-b)

Problem

sin(a+b)+sin(a−b)

Solution

1. Apply the sum identity for the sine function to the first term, sin(a+b)

sin(a+b)=sin(a)*cos(b)+cos(a)*sin(b)

2. Apply the difference identity for the sine function to the second term, sin(a−b)

sin(a−b)=sin(a)*cos(b)−cos(a)*sin(b)

3. Combine the expressions by adding the two expanded identities together.

sin(a)*cos(b)+cos(a)*sin(b)+sin(a)*cos(b)−cos(a)*sin(b)

4. Cancel the opposite terms cos(a)*sin(b) and −cos(a)*sin(b)

sin(a)*cos(b)+sin(a)*cos(b)

5. Combine like terms to reach the final simplified form.

2*sin(a)*cos(b)

Final Answer

sin(a+b)+sin(a−b)=2*sin(a)*cos(b)

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