if `siny=xsin(a+y) and sec^2y(dy)/(dx)=(A)/(1+x^(2)-2xcosa)` then the value of A is

Options

A. 2
B. cos a
C. sin a
D. none of these

Answer

To solve the problem, we start with the given equation:

\[ \sin y = x \sin(a + y) \]

### Step 1: Expand the right-hand side using the sine addition formula
Using the sine addition formula, we can expand \( \sin(a + y) \):

\[ \sin(a + y) = \sin a \cos y + \cos a \sin y \]

Substituting this into the equation gives:

\[ \sin y = x (\sin a \cos y + \cos a \sin y) \]

### Step 2: Rearrange the equation
Rearranging the equation, we have:

\[ \sin y = x \sin a \cos y + x \cos a \sin y \]

Now, we can move all terms involving \( \sin y \) to one side:

\[ \sin y - x \cos a \sin y = x \sin a \cos y \]

Factoring out \( \sin y \):

\[ \sin y (1 - x \cos a) = x \sin a \cos y \]

### Step 3: Isolate \( \sin y \)
Now, we can isolate \( \sin y \):

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

### Step 4: Find \( \tan y \)
To find \( \tan y \), we can use the identity \( \tan y = \frac{\sin y}{\cos y} \):

\[ \tan y = \frac{x \sin a}{1 - x \cos a} \]

### Step 5: Differentiate both sides with respect to \( x \)
Now we differentiate both sides with respect to \( x \):

Using the quotient rule on the right-hand side:

\[ \frac{d}{dx}(\tan y) = \sec^2 y \frac{dy}{dx} \]

Differentiating the right-hand side:

\[ \frac{d}{dx}\left(\frac{x \sin a}{1 - x \cos a}\right) = \frac{(1 - x \cos a)(\sin a) - (x \sin a)(-\cos a)}{(1 - x \cos a)^2} \]

This simplifies to:

\[ \frac{\sin a (1 - x \cos a) + x \sin a \cos a}{(1 - x \cos a)^2} = \frac{\sin a}{(1 - x \cos a)^2} \]

### Step 6: Set the derivatives equal
Now we equate the two derivatives:

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

### Step 7: Solve for \( A \)
From the original problem, we have:

\[ \sec^2 y \frac{dy}{dx} = \frac{A}{1 + x^2 - 2x \cos a} \]

Equating the two expressions gives:

\[ \frac{\sin a}{(1 - x \cos a)^2} = \frac{A}{1 + x^2 - 2x \cos a} \]

### Step 8: Cross multiply and simplify
Cross-multiplying results in:

\[ A(1 - x \cos a)^2 = \sin a (1 + x^2 - 2x \cos a) \]

### Step 9: Expand both sides
Expanding both sides:

Left-hand side:

\[ A(1 - 2x \cos a + x^2 \cos^2 a) \]

Right-hand side:

\[ \sin a + \sin a x^2 - 2 \sin a x \cos a \]

### Step 10: Compare coefficients
To find \( A \), we can compare coefficients of \( x^2 \):

From the left side, the coefficient of \( x^2 \) is \( A \cos^2 a \).

From the right side, the coefficient of \( x^2 \) is \( \sin a \).

Thus, we have:

\[ A \cos^2 a = \sin a \]

### Step 11: Solve for \( A \)
Finally, solving for \( A \):

\[ A = \frac{\sin a}{\cos^2 a} = \tan a \sec a \]

### Conclusion
The value of \( A \) is:

\[ A = \tan a \sec a \]