To simplify \( \left(\frac{1}{\sin^2 A} - 1\right) \), we start by using a trigonometric identity. Recognize that:

\(\frac{1}{\sin^2 A} = \csc^2 A\)

Thus, the expression becomes:

\(\csc^2 A - 1\)

Using the identity \(\csc^2 A = 1 + \cot^2 A\), substitute to get:

\((1 + \cot^2 A) - 1 = \cot^2 A\)

The expression simplifies to \(\cot^2 A\), but we need to evaluate whether it ultimately leads to one of the provided options.

We aim to express the original expression in terms of other trigonometric functions. Express \(\frac{1}{\sin^2 A} - 1\) in terms of cosine:

Using the Pythagorean identity \(\sin^2 A + \cos^2 A = 1\), we have:

\(\sin^2 A = 1 - \cos^2 A\)

Substituting into the original expression gives:

\(\frac{1}{\sin^2 A} = \frac{1}{1-\cos^2 A} = \frac{1}{\sin^2 A}\)

And simplifying further:

\(\frac{\cos^2 A}{\sin^2 A} = \cot^2 A\)

Alternatively, express \(\frac{1}{\sin^2 A} - 1\) as:

\(\frac{\cos^2 A}{\sin^2 A} = \cot^2 A = \tan^2 A\)

Thus, the correct answer is:

\(\tan^2 A\)