Find `dy/dx if x-siny=cosy`

Answer

To find \(\frac{dy}{dx}\) for the equation \(x - \sin y = \cos y\), we will differentiate both sides with respect to \(x\).

### Step-by-Step Solution:

1. **Start with the given equation:**
\[ x - \sin y = \cos y \]

2. **Differentiate both sides with respect to \(x\):**
- The derivative of \(x\) with respect to \(x\) is \(1\).
- For \(-\sin y\), we use the chain rule:
\[ -\frac{d}{dx}(\sin y) = -\cos y \cdot \frac{dy}{dx} \]
- For \(\cos y\), again using the chain rule:
\[ \frac{d}{dx}(\cos y) = -\sin y \cdot \frac{dy}{dx} \]

Putting this together, we differentiate the left-hand side and the right-hand side:
\[ 1 - \cos y \cdot \frac{dy}{dx} = -\sin y \cdot \frac{dy}{dx} \]

3. **Rearranging the equation:**
- Move all terms involving \(\frac{dy}{dx}\) to one side:
\[ -\cos y \cdot \frac{dy}{dx} + \sin y \cdot \frac{dy}{dx} = -1 \]
- Factor out \(\frac{dy}{dx}\):
\[ \frac{dy}{dx}(\sin y - \cos y) = -1 \]

4. **Solve for \(\frac{dy}{dx}\):**
\[ \frac{dy}{dx} = \frac{-1}{\sin y - \cos y} \]

### Final Answer:
\[ \frac{dy}{dx} = \frac{-1}{\sin y - \cos y} \]