Can you help me to establish identity of csc^2u-csc u sec u=cot^2 u

This is not a valid identity.

csc^{2}(u)-csc(u)sec(u)=cot^{2}(u)


Even solving as an equation it has no solutions in the reals.

Are you sure there is no typo?

There are a few identities to keep in mind.

csc^{2}(x)-1=cot^{2}(x)

By the given problem, then csc(x)sec(x)=1.

This is not true regardless of the value of x.

Here is a graph of cot^{2}(x) and csc^{2}(x)-csc(x)sec(x).

If they were equal, then the graph of each would overlap and be the same. They are not.