Verify the Identity sin(a+B)+sin(a-B)=2sin(a)cos(B)

Problem

sin(α+β)+sin(α−β)=2*sin(α)*cos(β)

Solution

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

sin(α+β)=sin(α)*cos(β)+cos(α)*sin(β)

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

sin(α−β)=sin(α)*cos(β)−cos(α)*sin(β)

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

sin(α+β)+sin(α−β)=(sin(α)*cos(β)+cos(α)*sin(β))+(sin(α)*cos(β)−cos(α)*sin(β))

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

sin(α+β)+sin(α−β)=sin(α)*cos(β)+sin(α)*cos(β)

5. Simplify the expression to reach the right side of the identity.

sin(α+β)+sin(α−β)=2*sin(α)*cos(β)

Final Answer

sin(α+β)+sin(α−β)=2*sin(α)*cos(β)

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