If `y=xsin(a+y),`prove that `(dy)/(dx)=(sin^2(a+y))/(sin(a+y)-ycos(a+y))dot`

Explanation

We have, `y=x sin (a+y)`


Differentiating with respect to `x` using chain rule,



`frac{d y}{d x}=x frac{d}{d x} sin (a+y)+sin (a+y) frac{d}{d x}(x) `


`Rightarrow frac{d y}{d x}=x cos (a+y) frac{d y}{d x}+sin (a+y)`


`Rightarrow{1-x cos (a+y)} frac{d y}{d x}=sin (a+y) `


`Rightarrow frac{d y}{d x}=frac{sin (a+y)}{1-x cos (a+y)} `


`Rightarrow frac{d y}{d x}=frac{sin (a+y)}{1-frac{y}{sin (a+y)} cos (a+y)} ldots ldots ldots ldots[frac{y}{sin (a+y)}=x] `


...