If $y=\frac{{\sin }^{2}x}{1+\cot x}+\frac{{\cos }^{2}x}{1+\tan x}$ then $y^1\left(x\right)$ is equal to

$y=\frac{{\sin }^{2}x}{1+\cot x}+\frac{{\cos }^{2}x}{1+\tan x}$

$y=\frac{{\sin }^{3}x}{\sin x+\cos x}+\frac{{\cos }^{3}x}{\cos x+\sin x}$

$y=\frac{{\sin }^{3}x+{\cos }^{3}x}{\sin x+\cos x}$

$y={\sin }^{2}x+{\cos }^{2}x-\sin x\cos x=1-1/2\sin 2x\text{ then }$

$y^{\mathrm{\prime }}\left(x\right)=-\cos 2x$