Verifying/proving of the trigonometric identities

I looked upon it, but I think that it's something wrong with the exercise.

\frac{\cos x}{\sec x} - \frac{\cos x}{\tan ^2 x}=\cot ^2 x

\sec x= \frac{1}{\cos x}

\frac{\cos x}{{1}{\cos x}} - \frac{\cos x}{{\sin ^2 x}{\cos ^2 x}}=

\cos ^2 x - \frac{\cos ^3 x}{\sin ^2 x}=

\frac{\cos ^2 x \sin ^2 x - \cos ^3 x}{\sin ^2 x}=

\frac{\cos ^2 x (\sin ^2 x - \cos x)}{\sin ^2 x}=

\frac{\cos ^2 x}{\sin ^2 x}= \cot ^2 x

\cot ^2 x (\sin ^2 - \cos x)

if there had been \sin ^ 2 x + \cos ^2 x instead of \sin ^2 x - \cos x it would have been true, but maybe I did some mistakes. On the whole, the idea is to use the formulas and calculate.

This is not a valid identity.

I also came to that answer and didn't get a valid identity, I think there is something wrong in the given, thanks anyway!