To solve the given trigonometric expression:

Expression:

$\frac{\tan\left(\alpha - \frac{\pi}{2}\right) - \cot\left(\pi - \alpha\right) + \cos\left(\alpha + \frac{3\pi}{2}\right)}{\sin\left(\pi - \alpha\right)}$

Step-by-Step Solution:

1. Simplify each term:

   • $\tan\left(\alpha - \frac{\pi}{2}\right)$: Using the identity $\tan(x - \frac{\pi}{2}) = -\cot(x)$, $\tan\left(\alpha - \frac{\pi}{2}\right) = -\cot(\alpha).$

   • $\cot\left(\pi - \alpha\right)$: Using the identity $\cot(\pi - x) = -\cot(x)$, $\cot\left(\pi - \alpha\right) = -\cot(\alpha).$

   • $\cos\left(\alpha + \frac{3\pi}{2}\right)$: Using the identity $\cos(x + 2\pi) = \cos(x)$ and $\cos(x + \frac{\pi}{2}) = -\sin(x)$, $\cos\left(\alpha + \frac{3\pi}{2}\right) = -\sin(\alpha).$

   • $\sin\left(\pi - \alpha\right)$: Using the identity $\sin(\pi - x) = \sin(x)$, $\sin\left(\pi - \alpha\right) = \sin(\alpha).$

2. Substitute these simplified terms into the expression: Substituting back: $\frac{\tan\left(\alpha - \frac{\pi}{2}\right) - \cot\left(\pi - \alpha\right) + \cos\left(\alpha + \frac{3\pi}{2}\right)}{\sin\left(\pi - \alpha\right)}$ becomes: $\frac{-\cot(\alpha) - (-\cot(\alpha)) - \sin(\alpha)}{\sin(\alpha)}.$

3. Simplify the numerator: Combine terms in the numerator: $-\cot(\alpha) + \cot(\alpha) - \sin(\alpha) = -\sin(\alpha).$

   So the expression simplifies to: $\frac{-\sin(\alpha)}{\sin(\alpha)}.$

4. Simplify the fraction: $\frac{-\sin(\alpha)}{\sin(\alpha)} = -1, \text{ provided } \sin(\alpha) \neq 0.$

Final Answer:

$\boxed{-1}, \text{ for } \sin(\alpha) \neq 0.$

Let me know if you'd like a deeper explanation or have further questions! Here are five related questions:

1. What happens if $\sin(\alpha) = 0$ in the original expression?
2. Can we generalize this type of trigonometric simplification?
3. What are the geometric interpretations of these trigonometric identities?
4. How would the result change if the numerator involved different trigonometric functions?
5. How does the periodicity of trigonometric functions affect this calculation?

Tip:

Always check for domain restrictions (like $\sin(\alpha) \neq 0$) when simplifying trigonometric expressions!