If `y=sqrt(sin x+sqrt(sin x)sqrt(sin x+...........oo)),` then `(2y-1) (dy)/(dx)` is equal to

Options

A. sin x
B. `-sin x`
C. `cos x`
D. `-cos x`

Answer

To solve the problem, we need to find the value of \((2y - 1) \frac{dy}{dx}\) given that

\[ y = \sqrt{\sin x + \sqrt{\sin x + \sqrt{\sin x + \ldots}}} \]

### Step 1: Set up the equation
Since the expression under the square root is the same as \(y\), we can write:

\[ y = \sqrt{\sin x + y} \]

### Step 2: Square both sides
Now, squaring both sides gives:

\[ y^2 = \sin x + y \]

### Step 3: Rearrange the equation
Rearranging the equation, we get:

\[ y^2 - y - \sin x = 0 \]

### Step 4: Differentiate both sides
Now we differentiate both sides with respect to \(x\):

\[ \frac{d}{dx}(y^2) - \frac{d}{dx}(y) - \frac{d}{dx}(\sin x) = 0 \]

Using the chain rule, we have:

\[ 2y \frac{dy}{dx} - \frac{dy}{dx} - \cos x = 0 \]

### Step 5: Factor out \(\frac{dy}{dx}\)
Now, we can factor out \(\frac{dy}{dx}\):

\[ (2y - 1) \frac{dy}{dx} = \cos x \]

### Step 6: Solve for \((2y - 1) \frac{dy}{dx}\)
Thus, we find:

\[ (2y - 1) \frac{dy}{dx} = \cos x \]

### Final Answer
So, the value of \((2y - 1) \frac{dy}{dx}\) is:

\[ \cos x \]