[tex]\dfrac{\cot x}{\csc x-\sin x}=\dfrac{\frac{\cos x}{\sin x}}{\frac{1}{\sin x}-\sin x}=
\dfrac{\frac{\cos x}{\sin x}}{\frac{1}{\sin x}-\frac{\sin^2x}{\sin x}}=
\dfrac{\frac{\cos x}{\sin x}}{\frac{1-\sin^2x}{\sin x}}=\\\\\\=
\dfrac{\frac{\cos x}{\sin x}}{\frac{\cos^2x}{\sin x}}=\dfrac{\cos x\cdot\sin x}{\sin x\cdot\cos^2x}=\dfrac{1}{\cos x}=\boxed{\sec x}[/tex]
