The problem requires verifying the trigonometric identity:

$\frac{\cos^2 \alpha + 1}{\sin \alpha} = \sin \alpha$

To verify this, let's analyze and simplify the left-hand side (LHS) and see if it can be transformed to match the right-hand side (RHS).

Step 1: Start with the LHS

$\frac{\cos^2 \alpha + 1}{\sin \alpha}$

Step 2: Apply the Pythagorean Identity

Notice that $\cos^2 \alpha + \sin^2 \alpha = 1$, so we can rewrite $\cos^2 \alpha$ as $1 - \sin^2 \alpha$: $\cos^2 \alpha + 1 = (1 - \sin^2 \alpha) + 1 = 2 - \sin^2 \alpha$

So, we substitute this into the LHS: $\frac{2 - \sin^2 \alpha}{\sin \alpha}$

Step 3: Break Down the Fraction

Rewrite $\frac{2 - \sin^2 \alpha}{\sin \alpha}$ as two separate fractions: $\frac{2}{\sin \alpha} - \frac{\sin^2 \alpha}{\sin \alpha}$

Simplify each term: $= \frac{2}{\sin \alpha} - \sin \alpha$

Step 4: Analyze Further

At this stage, we have an expression that doesn’t immediately simplify to $\sin \alpha$ directly. This suggests that the original identity may not be correct as stated, unless there is an adjustment or additional context needed.

Let me know if you'd like to go over another example, or if there's another part of the problem to clarify.

Do you want further details or have any questions?

Here are 5 additional questions for extended practice:

1. How do you verify identities in trigonometry effectively?
2. What are common Pythagorean identities in trigonometry?
3. How can you manipulate fractions involving trigonometric functions?
4. What strategies help when simplifying complex trigonometric expressions?
5. What are some common mistakes to avoid in trigonometric identities?

Tip: Always start with the more complex side of an identity and try to simplify it to match the other side.