To solve the expression \(\tan(2\alpha - 90^\circ)\), we can use the tangent subtraction identity and the co-function identity for tangent. Here are the steps:

### Step-by-Step Solution

1. **Rewrite the expression using the co-function identity**:
   The tangent co-function identity states:
   \[
   \tan(90^\circ - x) = \cot(x)
   \]

   Using this identity, we can rewrite \(\tan(2\alpha - 90^\circ)\) as:
   \[
   \tan(2\alpha - 90^\circ) = -\cot(2\alpha)
   \]

2. **Express \(\cot(2\alpha)\) in terms of tangent**:
   Recall that the cotangent function is the reciprocal of the tangent function:
   \[
   \cot(2\alpha) = \frac{1}{\tan(2\alpha)}
   \]

3. **Combine the steps**:
   Substituting this back into the equation gives us:
   \[
   \tan(2\alpha - 90^\circ) = -\frac{1}{\tan(2\alpha)}
   \]

So, the expression \(\tan(2\alpha - 90^\circ)\) simplifies to \(-\frac{1}{\tan(2\alpha)}\).

### Summary

\[
\tan(2\alpha - 90^\circ) = -\cot(2\alpha) = -\frac{1}{\tan(2\alpha)}
\]

Would you like more details or have any questions?

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Here are 5 related questions you might be interested in:

1. What is the value of \(\tan(2\alpha + 90^\circ)\)?
2. How does the tangent function behave as it approaches \(90^\circ\)?
3. What are the key identities for tangent and cotangent?
4. How can we derive the tangent addition and subtraction formulas?
5. How do you simplify expressions involving multiple trigonometric identities?

**Tip**: Understanding trigonometric identities, like those for tangent and cotangent, can help simplify complex trigonometric expressions and solve equations more easily.

Learn how to solve the trigonometric expression tan(2α - 90°) using tangent and cotangent identities. This detailed solution covers the tangent subtraction identity and the co-function identity, essential for advanced high school students studying trigonometry.

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