If $\displaystyle
siny = xsin(a+y)
$ and $\displaystyle
\frac{dy}{dx} = K \frac{sin^2y}{x^2}
$ , find K
I'm assuming that a is constant. If we expand your sine relationship and divide by x then $\displaystyle \frac{\sin y}{ x} = \sin a \cos y + \cos a \sin y$ then divide by $\displaystyle \sin y$ so
$\displaystyle \sin a \cot y + \cos a = \frac{1}{x} $ and upon differentiation gives $\displaystyle - \sin a \csc^2 y y' = - \frac{1}{x^2}$ which solving for y ' gives
$\displaystyle y' = \frac{1}{\sin a} \frac{\sin^2 y}{x^2}$
in which we can identify K.