Write each expression in terms of sine and cosine, and then simplify so that no quotients appear in the final expression and all functions are of $\theta$ only.
$$\frac{{\csc }^2(-\theta )-1}{1-{\cos }^2(-\theta )}$$
$$\frac{{\csc }^2(-\theta )-1}{1-{\cos }^2(-\theta )}=$$

Step 1 ：Write each expression in terms of sine and cosine, and then simplify so that no quotients appear in the final expression and all functions are of $\theta$ only.

Step 2 ：Given expression: $$\frac{{\csc }^2(-\theta )-1}{1-{\cos }^2(-\theta )}$$

Step 3 ：Convert the given expression into terms of sine and cosine. The cosecant function is the reciprocal of the sine function, so ${\csc }^2(-\theta )$ can be written as $\frac{1}{{\sin }^2(-\theta )}$. The denominator $1-{\cos }^2(-\theta )$ is equivalent to ${\sin }^2(-\theta )$ by the Pythagorean identity. Therefore, the expression can be simplified to $\frac{1}{{\sin }^2(-\theta )}-1$ divided by ${\sin }^2(-\theta )$.

Step 4 ：Simplify the expression to $\frac{{\cos }^2(-\theta )}{{\sin }^4(-\theta )}$. However, this expression still contains a quotient.

Step 5 ：To remove the quotient, we can multiply the numerator and the denominator by ${\sin }^4(-\theta )$, which gives ${\cos }^2(-\theta )$.

Step 6 ：The final simplified expression with no quotients and all functions of $\theta$ only is $\boxed{{\displaystyle {\cos }^2(-\theta )}}$.