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

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] `

`Rightarrow frac{d y}{d x}=frac{sin ^2(a+y)}{sin (a+y)-y cos (a+y)}`