To simplify \(\csc(90^\circ + a) / \tan(90^\circ - a)\), we’ll use trigonometric identities.

### Step 1: Apply Identities
1. **For \(\csc(90^\circ + a)\):**  
   \(\csc(90^\circ + a) = \sec(a)\)

2. **For \(\tan(90^\circ - a)\):**  
   \(\tan(90^\circ - a) = \cot(a)\)

### Step 2: Substitute and Simplify
\[
\frac{\csc(90^\circ + a)}{\tan(90^\circ - a)} = \frac{\sec(a)}{\cot(a)}
\]

### Step 3: Rewrite in Terms of Basic Trig Functions
1. **Rewrite \(\sec(a)\):**  
   \(\sec(a) = \frac{1}{\cos(a)}\)

2. **Rewrite \(\cot(a)\):**  
   \(\cot(a) = \frac{\cos(a)}{\sin(a)}\)

So:
\[
\frac{\sec(a)}{\cot(a)} = \frac{\frac{1}{\cos(a)}}{\frac{\cos(a)}{\sin(a)}} = \frac{1}{\cos(a)} \cdot \frac{\sin(a)}{\cos(a)} = \frac{\sin(a)}{\cos^2(a)}
\]

### Final Answer
The simplified form is:
\[
\frac{\sin(a)}{\cos^2(a)}
\]

Would you like a more detailed explanation or further assistance?

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**Related Questions:**

1. How can we simplify other trigonometric expressions using angle addition or subtraction?
2. What are other ways to express \(\csc(90^\circ + a)\) and \(\tan(90^\circ - a)\)?
3. How do we derive the trigonometric identities for angle transformations?
4. What are practical applications of trigonometric simplifications in calculus?
5. How do cofunction identities relate to transformations like \(\csc(90^\circ + a)\)?

**Tip:** Remember, transforming angles by \(90^\circ\), \(180^\circ\), or \(270^\circ\) often allows trigonometric expressions to be rewritten in simpler forms using cofunction identities.

Explore the simplification of the trigonometric expression cosec(90° + a) / tan(90° - a) with a clear, step-by-step explanation. This solution uses cofunction identities and angle transformation principles, ideal for high school students in Grades 10-12.

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