Step 1: Differentiate the given equation

Given the equation $y = \sin(x+y)$. Differentiating both sides with respect to $x$ using the chain rule:

Step 2: Apply the condition

We are given that $\frac{dy}{dx} = 0$. Substituting this value into the differentiated equation:

Step 3: Relate to the original equation

Using the trigonometric identity $\sin^2\theta + \cos^2\theta = 1$, we know that if $\cos(x+y) = 0$, then $\sin^2(x+y) = 1$.

Since $y = \sin(x+y)$, squaring both sides gives $y^2 = \sin^2(x+y)$.

Therefore, $y^2 = 1$, which implies $y = \pm 1$.