If sin y = xsin(a+y) , then displaystyledy/dx is:

The correct answer is Option (4) → \(\displaystyle\frac{\sin^2(a+y)}{\sin a}\)

Since x is easy to isolate, x can be written in terms of y:

    \( \displaystyle x = \frac{\sin y}{\sin(a+y)} \)

Differentiating with respect to y by the quotient rule:

    \( \displaystyle \frac{dx}{dy} = \frac{\sin(a+y)\cos y - \sin y\cos(a+y)}{\sin^2(a+y)} \)

The numerator is the expansion of \(\sin(A - B)\) with A = a + y and B = y:

    \( \sin(a+y)\cos y - \cos(a+y)\sin y = \sin(a+y-y) = \sin a \)

So

    \( \displaystyle \frac{dx}{dy} = \frac{\sin a}{\sin^2(a+y)} \)

    \( \displaystyle \frac{dy}{dx} = \frac{1}{dx/dy} = \frac{\sin^2(a+y)}{\sin a} \)