Based on Pythagorean identities, the correct equation from the options given is:

$\sec^2 \theta - \tan^2 \theta = 1$.

This equation is a well-known Pythagorean identity, where:

$\sec^2 \theta = 1 + \tan^2 \theta$

Let's check each option:

1. $\sin^2 \theta - 1 = \cos^2 \theta$
   This is incorrect because $\sin^2 \theta + \cos^2 \theta = 1$, not $\sin^2 \theta - 1 = \cos^2 \theta$.

2. $\sec^2 \theta - \tan^2 \theta = -1$
   This is incorrect, because $\sec^2 \theta - \tan^2 \theta = 1$.

3. $-\cos^2 \theta - 1 = -\sin^2 \theta$
   This is incorrect because simplifying $-\cos^2 \theta - 1$ doesn't yield $-\sin^2 \theta$. Instead, $\cos^2 \theta + \sin^2 \theta = 1$.

4. $\cot^2 \theta - \csc^2 \theta = -1$
   This is correct. It follows from the identity $\csc^2 \theta = 1 + \cot^2 \theta$, which rearranges to $\cot^2 \theta - \csc^2 \theta = -1$.

Thus, the correct equation is:
$\sec^2 \theta - \tan^2 \theta = 1$.

Would you like details on how each Pythagorean identity is derived?

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Here are 5 related questions:

1. How do the Pythagorean identities relate to the unit circle?
2. What are the reciprocal identities in trigonometry?
3. How can the Pythagorean identities be used to simplify trigonometric expressions?
4. What is the connection between Pythagorean identities and the Pythagorean theorem?
5. How do we prove $\sec^2 \theta - \tan^2 \theta = 1$ geometrically?

Tip: Always remember that each trigonometric function has a corresponding reciprocal identity (e.g., $\sec \theta = \frac{1}{\cos \theta}$).