The question in the image is a trigonometric expression. The problem states:

• Given that $\tan \alpha^\circ = -$ (implying $\tan \alpha^\circ$ is negative) and $\alpha^\circ$ is in the second quadrant.
• We need to calculate the following expression:

$\frac{\sin(90^\circ - \alpha^\circ) - \cos(180^\circ - \alpha^\circ)}{\tan(270^\circ + \alpha^\circ) + \cot(360^\circ - \alpha^\circ)}$

Step-by-step solution:

1. Identify trigonometric identities in terms of $\alpha^\circ$:

   • $\sin(90^\circ - \alpha^\circ) = \cos(\alpha^\circ)$ (co-function identity)
   • $\cos(180^\circ - \alpha^\circ) = -\cos(\alpha^\circ)$ (because $\cos$ is negative in the second quadrant)
   • $\tan(270^\circ + \alpha^\circ) = \cot(\alpha^\circ)$ (since $\tan(270^\circ + x) = \cot(x)$)
   • $\cot(360^\circ - \alpha^\circ) = \cot(\alpha^\circ)$ (as $\cot$ repeats every 180 degrees)

2. Substitute the identities: The expression simplifies to:

   $\frac{\cos(\alpha^\circ) - (-\cos(\alpha^\circ))}{\cot(\alpha^\circ) + \cot(\alpha^\circ)}$

   This simplifies to:

   $\frac{\cos(\alpha^\circ) + \cos(\alpha^\circ)}{2 \cdot \cot(\alpha^\circ)}$

   Which further simplifies to:

   $\frac{2\cos(\alpha^\circ)}{2 \cdot \cot(\alpha^\circ)} = \frac{\cos(\alpha^\circ)}{\cot(\alpha^\circ)}$

3. Simplify the final result: Since $\frac{\cos(\alpha^\circ)}{\cot(\alpha^\circ)} = \sin(\alpha^\circ)$, the final answer is:

   $\sin(\alpha^\circ)$

Would you like further clarification on this solution or details on any specific steps?

Here are 5 follow-up questions that could help expand your understanding:

1. Why is $\tan \alpha^\circ$ negative in the second quadrant?
2. Can you explain how the co-function identities are derived?
3. How does the periodicity of trigonometric functions help simplify expressions?
4. What is the relationship between $\tan$ and $\cot$?
5. How would this problem change if $\alpha^\circ$ were in a different quadrant?

Tip: Always verify the quadrant of an angle when solving trigonometric problems to ensure correct sign usage for sine, cosine, and tangent.