If ` sin y=x cos ( a + y) ,` then ` (dy)/(dx) =`

Options

A. `(sin^2 (a+y))/(sin a)`
B. `( cos a)/( cos^2 (a +y))`
C. `(cos^2 (a+y))/(cos a)`
D. None

Answer

To solve the problem \( \sin y = x \cos(a + y) \) and find \( \frac{dy}{dx} \), we will use implicit differentiation.

### Step-by-Step Solution:

1. **Differentiate both sides with respect to \( x \)**:
\[ \frac{d}{dx}(\sin y) = \frac{d}{dx}(x \cos(a + y)) \]

2. **Apply the chain rule on the left side**:
\[ \cos y \frac{dy}{dx} = \frac{d}{dx}(x \cos(a + y)) \]

3. **Use the product rule on the right side**:
\[ \cos y \frac{dy}{dx} = \cos(a + y) + x \frac{d}{dx}(\cos(a + y)) \]

4. **Differentiate \( \cos(a + y) \) using the chain rule**:
\[ \frac{d}{dx}(\cos(a + y)) = -\sin(a + y) \frac{dy}{dx} \]
Thus, we can rewrite the equation as:
\[ \cos y \frac{dy}{dx} = \cos(a + y) - x \sin(a + y) \frac{dy}{dx} \]

5. **Rearranging the equation**:
\[ \cos y \frac{dy}{dx} + x \sin(a + y) \frac{dy}{dx} = \cos(a + y) \]
Factor out \( \frac{dy}{dx} \):
\[ \left(\cos y + x \sin(a + y)\right) \frac{dy}{dx} = \cos(a + y) \]

6. **Solve for \( \frac{dy}{dx} \)**:
\[ \frac{dy}{dx} = \frac{\cos(a + y)}{\cos y + x \sin(a + y)} \]

### Final Answer:
\[ \frac{dy}{dx} = \frac{\cos(a + y)}{\cos y + x \sin(a + y)} \]