Help with my trig homework: Trig Identities?

That is not actually an identity at all. The word should not be used so loosely.

For disproof by counterexample, let x = 0, and see how an undefined expression is equated with the real number 1.

The purported proof fails on the final step:

The expression [1/sin²(x)]sin²(x) is equal to 1 only if sin²(x) ≠ 0.

1) Reciprocal identity (csc(x) = 1/sin(x)).

2) Pythagorean identity (cos^2(x) + sin^2(x) = 1, and any arrangements of this thereof when pit in terms of cos^2(x) or sin^2(x)).

3) Algebra (a nonzero expression/number times its reciprocal is always 1).

reciprocal (csc and sin are reciprocals)

algebra (cos^2 x + sin^2 x = 1; so sin^2 x = 1 - cos^2 x)

quotient? or algebra again (not sure on the rule descriptions and if you can use a rule twice; 1/x * x = 1)