Let's simplify the given expression step by step:

The expression is:
$\frac{\sin x (1 + \sin x)}{1 - \cos^2 x}$

1. Use the Pythagorean identity:
   Recall that $\sin^2 x + \cos^2 x = 1$, so we can rewrite the denominator: $1 - \cos^2 x = \sin^2 x$ Therefore, the expression becomes: $\frac{\sin x (1 + \sin x)}{\sin^2 x}$

2. Simplify:
   We can now simplify the fraction: $\frac{\sin x (1 + \sin x)}{\sin^2 x} = \frac{\sin x}{\sin^2 x} \cdot (1 + \sin x)$ $= \frac{1 + \sin x}{\sin x}$

Thus, the simplified expression is: $\frac{1 + \sin x}{\sin x}$

This is the final, simplified form of the given expression.

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Would you like a more detailed breakdown or help with any further steps? Here are some follow-up questions you could explore:

1. How do you simplify trigonometric expressions involving other identities, like $\tan x$ or $\sec x$?
2. How can we verify the identity with specific values of $x$?
3. What would be the behavior of this expression at certain values of $x$, such as $x = 0$ or $x = \pi$?
4. How do we handle limits of trigonometric expressions like this when approaching undefined points?
5. Can this expression be used in solving real-world problems, such as wave functions or oscillations?

Tip: Always look for trigonometric identities (like $\sin^2 x + \cos^2 x = 1$) when simplifying expressions to make the process easier!