If `2 sin x. cos y=1, ` then `(d ^(2)y)/(dx ^(2)) at ((pi)/(4), (pi)/(4))` is …….

Options

A. `-4`
B. `-2`
C. `-6`
D. 0

Answer

To solve the problem, we need to find the second derivative of \( y \) with respect to \( x \) at the point \( \left( \frac{\pi}{4}, \frac{\pi}{4} \right) \) given the equation:

\[ 2 \sin x \cos y = 1 \]

### Step 1: Simplify the equation
We start by simplifying the given equation:

\[ \sin x \cos y = \frac{1}{2} \]

### Step 2: Differentiate both sides
Next, we differentiate both sides with respect to \( x \). Using the product rule on the left side, we have:

\[ \frac{d}{dx}(\sin x \cos y) = \cos x \cos y - \sin x \sin y \frac{dy}{dx} \]

The right side is a constant, so its derivative is 0:

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

### Step 3: Solve for \( \frac{dy}{dx} \)
Rearranging the equation gives:

\[ \sin x \sin y \frac{dy}{dx} = \cos x \cos y \]

Thus,

\[ \frac{dy}{dx} = \frac{\cos x \cos y}{\sin x \sin y} \]

This can be rewritten using cotangent:

\[ \frac{dy}{dx} = \cot x \cot y \]

### Step 4: Evaluate \( \frac{dy}{dx} \) at \( \left( \frac{\pi}{4}, \frac{\pi}{4} \right) \)
Now we substitute \( x = \frac{\pi}{4} \) and \( y = \frac{\pi}{4} \):

\[ \frac{dy}{dx} \bigg|_{\left( \frac{\pi}{4}, \frac{\pi}{4} \right)} = \cot \frac{\pi}{4} \cdot \cot \frac{\pi}{4} = 1 \cdot 1 = 1 \]

### Step 5: Differentiate \( \frac{dy}{dx} \) to find \( \frac{d^2y}{dx^2} \)
Next, we differentiate \( \frac{dy}{dx} \) again to find \( \frac{d^2y}{dx^2} \):

\[ \frac{d^2y}{dx^2} = \frac{d}{dx}(\cot x \cot y) \]

Using the product rule:

\[ \frac{d^2y}{dx^2} = -\csc^2 x \cot y + \cot x (-\csc^2 y \frac{dy}{dx}) \]

Substituting \( \frac{dy}{dx} = 1 \):

\[ \frac{d^2y}{dx^2} = -\csc^2 x \cot y - \cot x \csc^2 y \]

### Step 6: Evaluate \( \frac{d^2y}{dx^2} \) at \( \left( \frac{\pi}{4}, \frac{\pi}{4} \right) \)
Now we substitute \( x = \frac{\pi}{4} \) and \( y = \frac{\pi}{4} \):

\[ \frac{d^2y}{dx^2} \bigg|_{\left( \frac{\pi}{4}, \frac{\pi}{4} \right)} = -\csc^2 \frac{\pi}{4} \cdot 1 - 1 \cdot \csc^2 \frac{\pi}{4} \]

Since \( \csc \frac{\pi}{4} = \sqrt{2} \):

\[ \frac{d^2y}{dx^2} = -2 \cdot 2 = -4 \]

### Final Answer
Thus, the value of \( \frac{d^2y}{dx^2} \) at \( \left( \frac{\pi}{4}, \frac{\pi}{4} \right) \) is:

\[ \boxed{-4} \]