Answer
$$\sin(x+y)+\sin(x-y)=2\sin x\cos y$$
The equation has been verified to be an identity.
Work Step by Step
$$\sin(x+y)+\sin(x-y)=2\sin x\cos y$$
Let's examine the left side.
$$X=\sin(x+y)+\sin(x-y)$$
According to the sine sum and difference identities:
$$\sin(A+B)=\sin A\cos B+\cos A\sin B$$
$$\sin(A-B)=\sin A\cos B-\cos A\sin B$$
Therefore,
$$X=(\sin x\cos y+\cos y\sin x)+(\sin x\cos y-\cos y\sin x)$$
$$X=(\sin x\cos y+\sin x\cos y)+(\cos y\sin x-\cos y\sin x)$$
$$X=2\sin x\cos y$$
Thus 2 sides are equal. This equation has been verified to be an identity.
