If ` y + sin y = cos x, then (dy)/(dx)` is equal to

Options

A. `- (sin x)/(1 + cos y),y = (2n + 1) pi`
B. `(sin x)/(1 + cos y), y ne (2n + 1) pi`
C. `-(sin x)/(1 + cos y), y ne (2n +1) pi`
D. None of the above

Explanation

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

### Step-by-Step Solution:

1. **Differentiate Both Sides**:
We start with the equation:
\[ y + \sin y = \cos x \]
Now, differentiate both sides with respect to \(x\):
\[ \frac{d}{dx}(y) + \frac{d}{dx}(\sin y) = \frac{d}{dx}(\cos x) \]

2. **Apply the Chain Rule**:
The derivative of \(y\) with respect to \(x\) is \(\frac{dy}{dx}\).
For \(\sin y\), we apply the chain rule:
\[ \frac{d}{dx}(\sin y) = \cos y \cdot \frac{dy}{dx} \]
The derivative of \(\cos x\) is:
\[ \frac{d}{dx}(\cos x) = -\sin x \]
So, substituting these derivatives into our equation gives:
\[ \frac{dy}{dx} + \cos y \cdot \frac{dy}{dx} = -\sin x \]

3. **Factor Out \(\frac{dy}{dx}\)**:
We can factor \(\frac{dy}{dx}\) out from the left side:
\[ \frac{dy}{dx}(1 + \cos y) = -\sin x \]

4. **Solve for \(\frac{dy}{dx}\)**:
Now, we can isolate \(\frac{dy}{dx}\):
\[ \frac{dy}{dx} = \frac{-\sin x}{1 + \cos y} \]

### Final Result:
Thus, the derivative \(\frac{dy}{dx}\) is given by:
\[ \frac{dy}{dx} = \frac{-\sin x}{1 + \cos y} \]