Prove $$\frac{d^{n}}{dx^n}(\frac{x^n}{1+x^2})=n!\sin(y)\{\sin(y)-\dbinom{n}{1}\cos(y)\sin(2y)+\dbinom{n}{2}\cos^2(y)\sin(3y)-....\}, x=\cot(y)$$
I tried to do it by induction I got $$\frac{d^{n}}{dx^n}(\frac{x^n}{1+x^2})=\frac{d}{dx}(\frac{d^{n-1}}{dx^{n-1}}(\frac{x.x^{n-1}}{1+x^2}))=x\frac{d^{n}}{dx^{n}}(\frac{x^{n-1}}{1+x^2})+n\frac{d^{n-1}}{dx^{n}}(\frac{x^{n-1}}{1+x^2})$$
But I am not getting the result anyone please help/
Can anyone please give me a complete solution