Thank you for the clarification! The problem now reads:

\[
\text{If } (\tan A + \cot A)^2 = \text{?}
\]

Let's solve it.

### Step 1: Expand the square
\[
(\tan A + \cot A)^2 = \tan^2 A + 2 \tan A \cdot \cot A + \cot^2 A
\]

### Step 2: Simplify the expression
Recall that \(\cot A = \frac{1}{\tan A}\), so we have:
\[
(\tan A + \cot A)^2 = \tan^2 A + 2 \cdot \tan A \cdot \frac{1}{\tan A} + \frac{1}{\tan^2 A}
\]

Simplify further:
\[
= \tan^2 A + 2 + \frac{1}{\tan^2 A}
\]

### Step 3: Use the identity for \( \tan^2 A \)
Let \( x = \tan^2 A \). Then the expression becomes:
\[
x + 2 + \frac{1}{x}
\]

### Step 4: Common denominator (if needed)
If you need a single expression, you can write this as:
\[
\frac{x^2 + 2x + 1}{x} = \frac{(x + 1)^2}{x}
\]

But most likely, the answer to the original problem is:
\[
(\tan A + \cot A)^2 = \tan^2 A + 2 + \cot^2 A
\]

If the question asks for a particular value of \(A\), then we would need to substitute that value in.

Would you like more details or have any questions?

### Relative Questions:
1. What is the value of \( \tan A \) when \( A = 45^\circ \)?
2. How does the identity \( \tan^2 A + 1 = \sec^2 A \) relate to this problem?
3. What happens if \( \tan A = 1 \) in the original expression?
4. How can this problem be solved using trigonometric identities?
5. What is the geometric interpretation of \( \tan A \) and \( \cot A \)?

### Tip:
Understanding the relationship between \( \tan \) and \( \cot \) functions can greatly simplify trigonometric identities and expressions!

Explore how to solve the (tan A + cot A)^2 trigonometric identity problem step-by-step. This problem involves expanding and simplifying trigonometric expressions using fundamental identities and is suitable for high school students studying trigonometry.

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