The expression you've provided is:

$\lim_{x \to \left( \frac{\pi}{2} \right)^+} \frac{\cos(x)}{1 - \sin(x)}$

This represents the limit of the function $\frac{\cos(x)}{1 - \sin(x)}$ as $x$ approaches $\frac{\pi}{2}$ from the right-hand side (i.e., from values greater than $\frac{\pi}{2}$).

Step-by-step evaluation:

1. Behavior of $\cos(x)$ as $x \to \frac{\pi}{2}^+$:

   • The cosine function approaches $0$ as $x \to \frac{\pi}{2}^+$ (from the right side). Specifically, $\cos\left(\frac{\pi}{2}\right) = 0$.

2. Behavior of $1 - \sin(x)$ as $x \to \frac{\pi}{2}^+$:

   • The sine function approaches $1$ as $x \to \frac{\pi}{2}$. Thus, $1 - \sin(x)$ approaches $0$ as $x \to \frac{\pi}{2}^+$.

3. Right-hand limit:

   • When approaching from the right side, $\sin(x)$ is slightly greater than $1$ (since $\sin\left(\frac{\pi}{2}^+\right) > 1$), so $1 - \sin(x)$ is slightly negative. Therefore, the denominator becomes slightly negative as $x \to \frac{\pi}{2}^+$.

4. Form of the limit:

   • As $x \to \frac{\pi}{2}^+$, the numerator approaches $0$ (since $\cos(x) \to 0$), and the denominator approaches $0$ but negative (since $1 - \sin(x) \to 0^-$).

Thus, the limit takes the form $\frac{0}{0^-}$. This is an indeterminate form.

Result:

Since the numerator approaches $0$ and the denominator approaches $0$ negatively, the overall limit approaches $-\infty$.

$\boxed{-\infty}$

Would you like any more details on this, or do you have further questions?

Here are 5 related questions to expand on this topic:

1. What happens if you approach $\frac{\pi}{2}$ from the left side instead of the right?
2. How can L'Hôpital's Rule be used to evaluate indeterminate limits like this one?
3. Can you provide a geometric interpretation of the behavior of sine and cosine around $\frac{\pi}{2}$?
4. What are other common trigonometric limits involving $\sin(x)$ and $\cos(x)$?
5. How does the limit change if the denominator were $\cos(x)$ instead of $1 - \sin(x)$?

Tip: When encountering indeterminate forms like $\frac{0}{0}$, L'Hôpital's Rule is often useful to resolve the limit.