Simplify $\sin (x-y) \cos y + \cos (x-y) \sin y$.

$ \textbf{(A)}\ 1\qquad\textbf{(B)}\ \sin x\qquad\textbf{(C)}\ \cos x\qquad\textbf{(D)}\ \sin x \cos 2y\qquad\textbf{(E)}\ \cos x \cos 2y $
To simplify the expression \(\sin (x-y) \cos y + \cos (x-y) \sin y\), we can use the angle addition formulas for sine and cosine.

1. Recall the angle subtraction identities:
   \[
   \sin (x-y) = \sin x \cos y - \cos x \sin y
   \]
   \[
   \cos (x-y) = \cos x \cos y + \sin x \sin y
   \]

2. Substitute these identities into the given expression:
   \[
   \sin (x-y) \cos y + \cos (x-y) \sin y
   \]
   \[
   = (\sin x \cos y - \cos x \sin y) \cos y + (\cos x \cos y + \sin x \sin y) \sin y
   \]

3. Distribute \(\cos y\) and \(\sin y\) respectively:
   \[
   = \sin x \cos^2 y - \cos x \sin y \cos y + \cos x \cos y \sin y + \sin x \sin^2 y
   \]

4. Combine like terms:
   \[
   = \sin x \cos^2 y + \sin x \sin^2 y - \cos x \sin y \cos y + \cos x \cos y \sin y
   \]

5. Notice that the terms \(- \cos x \sin y \cos y\) and \(\cos x \cos y \sin y\) cancel each other out:
   \[
   = \sin x (\cos^2 y + \sin^2 y)
   \]

6. Use the Pythagorean identity \(\cos^2 y + \sin^2 y = 1\):
   \[
   = \sin x \cdot 1
   \]
   \[
   = \sin x
   \]

The final answer is \(\boxed{\sin x}\).