Question

is 1+cos^2x/sin^2x=1 an identity? and how would you prove that it is?

Answer 1

It is not an identity, but it is an equation with solutions.

Answer 2

For it to be identity, it has to be true for any value of x.

So first you can simplify it to \(1+\cot^2x=1\) Go further and you have \(\cot^2x=0\)
Just pick any value of x that make this equation false. Use it as counterexample.