Simplify by reducing to a single term :
`sin (x -y) cos x - cos (x-y) sin x.`

Answer

To simplify the expression \( \sin(x - y) \cos x - \cos(x - y) \sin x \), we can use the sine subtraction formula. Here’s a step-by-step solution:

### Step 1: Identify the expression
We start with the expression:
\[ \sin(x - y) \cos x - \cos(x - y) \sin x \]

### Step 2: Recognize the sine subtraction formula
Recall the sine subtraction formula:
\[ \sin(A - B) = \sin A \cos B - \cos A \sin B \]
In our case, we can let \( A = x - y \) and \( B = x \).

### Step 3: Rewrite the expression using the formula
Using the sine subtraction formula, we can rewrite our expression:
\[ \sin(x - y) \cos x - \cos(x - y) \sin x = \sin((x - y) - x) \]

### Step 4: Simplify the argument of the sine function
Now we simplify the argument:
\[ (x - y) - x = -y \]
Thus, we have:
\[ \sin((x - y) - x) = \sin(-y) \]

### Step 5: Use the property of sine
We know that:
\[ \sin(-\theta) = -\sin(\theta) \]
Applying this property, we get:
\[ \sin(-y) = -\sin y \]

### Final Answer
Therefore, the simplified expression is:
\[ -\sin y \]

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