trigonometry - Proving the identity $csc x−sin x = (cot x)(cos x)$

We recently started Trigonometry and I was trying to solve this.

$$\csc x − \sin x=(\cot x)(\cos x)$$

So starting with LHS:

$$\begin{align} \frac{1}{\sin x} − \sin x &= \frac{1 − (\sin x)^2}{ \sin x } \\ &= \frac{(\cos x)^2 }{ \sin x } \end{align}$$

I am stuck now and wanted to know how should I proceed. Is this much correct?

Answer

Its done. Just split the numerator as:

$$\frac{\cos x . \cos x }{\sin x}$$

$$= \frac{\cos x }{\sin x} . \cos x$$

$$= \cot x . \cos x$$

Proved!