[tex]\dfrac{\cot x}{1+\csc x}=\dfrac{\csc x-1}{\cot x}\\
\cot^2 x=(1+\csc x)(\csc x-1)\\
\cot^2 x=-(1+\csc x)(1-\csc x)\\
\cot^2 x=-(1-\csc^2 x)\\
\cot^2 x=-1+\csc^2 x\\
\left(\dfrac{\cos x}{\sin x}\right)^2=-1+\left(\dfrac{1}{\sin x}\right)^2\\
\dfrac{\cos^2 x}{\sin ^2x}=-1+\dfrac{1}{\sin^2 x}\\
\cos^2 x=-\sin^2x +1\\
\boxed{\sin^2 x+\cos ^2 x=1}[/tex]
