Let's simplify the given expression:

$\frac{\tan^2 \theta (\sec^2 \theta - 1)}{\sin^2 \theta}$

Step 1: Use the Trigonometric Identity

We know that:

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

Using this identity, we substitute:

$\frac{\tan^2 \theta (\tan^2 \theta)}{\sin^2 \theta}$

$= \frac{\tan^4 \theta}{\sin^2 \theta}$

Step 2: Express in Terms of Sine and Cosine

Since $\tan \theta = \frac{\sin \theta}{\cos \theta}$, we rewrite:

$\tan^4 \theta = \left(\frac{\sin \theta}{\cos \theta}\right)^4 = \frac{\sin^4 \theta}{\cos^4 \theta}$

Thus, our expression becomes:

$\frac{\frac{\sin^4 \theta}{\cos^4 \theta}}{\sin^2 \theta}$

Step 3: Simplify the Fraction

$= \frac{\sin^4 \theta}{\cos^4 \theta} \times \frac{1}{\sin^2 \theta}$

$= \frac{\sin^2 \theta}{\cos^4 \theta}$

$= \frac{\sin^2 \theta}{\cos^2 \theta \cdot \cos^2 \theta}$

Since $\frac{\sin^2 \theta}{\cos^2 \theta} = \tan^2 \theta$, we obtain:

$\frac{\tan^2 \theta}{\cos^2 \theta}$

Final Answer:

$\frac{\tan^2 \theta}{\cos^2 \theta}$

Let me know if you need further clarification!

Related Questions:

1. What are the fundamental Pythagorean identities in trigonometry?
2. How do you derive $\sec^2 \theta - 1 = \tan^2 \theta$?
3. Why is expressing trigonometric functions in terms of sine and cosine useful?
4. What is an alternative way to simplify this expression using a different approach?
5. How does the simplification change if a different trigonometric identity is used?

Tip: Always check for fundamental trigonometric identities when simplifying expressions!