The problem you've shared appears to have two parts:

1. The Given Condition:
   $\tan \theta = -3$

2. Expression to Simplify:
   $\frac{1 - \sin^2 \theta}{\cos^2 \theta}$

Step 1: Analyze the Condition

Using the Pythagorean identity: $1 - \sin^2 \theta = \cos^2 \theta$ Substitute this into the expression: $\frac{1 - \sin^2 \theta}{\cos^2 \theta} = \frac{\cos^2 \theta}{\cos^2 \theta}$

Step 2: Simplify the Expression

Since $\cos^2 \theta$ is the same in both the numerator and denominator, they cancel each other out: $\frac{\cos^2 \theta}{\cos^2 \theta} = 1$

Thus, the simplified form of the expression is: $\boxed{1}$

So, regardless of the value of $\tan \theta$, as long as $\cos \theta \neq 0$, the expression simplifies to $1$.

Would you like further details on any specific part of the solution?

Here are some related questions for deeper understanding:

1. How does the Pythagorean identity relate to trigonometric simplifications?
2. What would happen if $\tan \theta$ had a different value?
3. Can this type of simplification be used for other trigonometric identities?
4. What is the significance of the tangent function in this problem?
5. How does knowing the value of $\tan \theta$ affect the values of $\sin \theta$ and $\cos \theta$?

Tip: In trigonometric problems, try using fundamental identities like the Pythagorean identities to simplify expressions early on. It often reduces the complexity of the problem.