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

Options

A. `(sin a)/(sin^2 (a+y))`
B. `(sin^2(a+y))/(sin a )`
C. `sin a sin^2 (a+y)`
D. `(sin^2(a-y))/(sina)`

Answer

To solve the equation \( \sin y = x \sin(a + y) \) and find \( \frac{dy}{dx} \), we can follow these steps:

### Step 1: Differentiate both sides with respect to \( x \)
We start with the equation:
\[ \sin y = x \sin(a + y) \]
Now, we differentiate both sides with respect to \( x \):
\[ \frac{d}{dx}(\sin y) = \frac{d}{dx}(x \sin(a + y)) \]

### Step 2: Apply the chain rule on the left side
Using the chain rule on the left side:
\[ \cos y \frac{dy}{dx} = \sin(a + y) + x \cos(a + y) \frac{dy}{dx} \]

### Step 3: Rearrange the equation
Now, we can rearrange the equation to isolate \( \frac{dy}{dx} \):
\[ \cos y \frac{dy}{dx} - x \cos(a + y) \frac{dy}{dx} = \sin(a + y) \]
Factoring out \( \frac{dy}{dx} \):
\[ \left(\cos y - x \cos(a + y)\right) \frac{dy}{dx} = \sin(a + y) \]

### Step 4: Solve for \( \frac{dy}{dx} \)
Now, we can solve for \( \frac{dy}{dx} \):
\[ \frac{dy}{dx} = \frac{\sin(a + y)}{\cos y - x \cos(a + y)} \]

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