Cosec theta -sin theta =m cube sec theta - cos theta = n cube prove that m^4n^2+n^4m^2=1

This is false. (Is there any more information? Are you sure there is no typo?)

Let and . Then

In the first equation,

In the second,

So , and

so we're left to prove that

Now

and both and are non-negative, so

and cannot possibly be equal to 1.