To solve the expression \( A = ( \tan 23^\circ + \cot 23^\circ )^2 - ( \cos 23^\circ - \tan 23^\circ )^2 \), we can use trigonometric identities and algebraic manipulation.

First, recall the following trigonometric identities:
1. \(\cot \theta = \frac{1}{\tan \theta}\)
2. \(\tan^2 \theta + 1 = \sec^2 \theta\)
3. \(\sec \theta = \frac{1}{\cos \theta}\)

Let's start by simplifying each part of the expression.

### Simplifying \(( \tan 23^\circ + \cot 23^\circ )^2\):
\[
\tan 23^\circ + \cot 23^\circ = \tan 23^\circ + \frac{1}{\tan 23^\circ}
\]
Let \( x = \tan 23^\circ \), then:
\[
\tan 23^\circ + \cot 23^\circ = x + \frac{1}{x}
\]
\[
(x + \frac{1}{x})^2 = x^2 + 2 \cdot \frac{x}{x} + \frac{1}{x^2} = x^2 + 2 + \frac{1}{x^2}
\]

### Simplifying \(( \cos 23^\circ - \tan 23^\circ )^2\):
\[
(\cos 23^\circ - \tan 23^\circ)^2 = \cos^2 23^\circ - 2 \cos 23^\circ \tan 23^\circ + \tan^2 23^\circ
\]

Now, let's combine these results into the original expression:
\[
A = ( \tan 23^\circ + \cot 23^\circ )^2 - ( \cos 23^\circ - \tan 23^\circ )^2
\]
\[
A = \left( \tan 23^\circ + \frac{1}{\tan 23^\circ} \right)^2 - ( \cos^2 23^\circ - 2 \cos 23^\circ \tan 23^\circ + \tan^2 23^\circ )
\]
\[
A = \left( \tan^2 23^\circ + 2 + \frac{1}{\tan^2 23^\circ} \right) - \left( \cos^2 23^\circ - 2 \cos 23^\circ \tan 23^\circ + \tan^2 23^\circ \right)
\]

Notice that \(\cos^2 23^\circ\) can be written as \(1 - \sin^2 23^\circ\), but it doesn't simplify the expression significantly. Instead, let's focus on the terms involving \(\tan 23^\circ\).

Recall that \(\tan^2 23^\circ + 1 = \sec^2 23^\circ\), and \(\sec 23^\circ = \frac{1}{\cos 23^\circ}\).

We can rewrite the expression as:
\[
A = \left( \tan^2 23^\circ + 2 + \frac{1}{\tan^2 23^\circ} \right) - \left( \cos^2 23^\circ - 2 \cos 23^\circ \tan 23^\circ + \tan^2 23^\circ \right)
\]

Since \(\tan 23^\circ = \frac{\sin 23^\circ}{\cos 23^\circ}\), we can simplify further, but it becomes cumbersome. Instead, let's use a numerical approach to evaluate the expression directly.

Using a calculator:
\[
\tan 23^\circ \approx 0.4245
\]
\[
\cot 23^\circ \approx 2.3558
\]
\[
\cos 23^\circ \approx 0.9205
\]

Now, substitute these values into the expression:
\[
A = (0.4245 + 2.3558)^2 - (0.9205 - 0.4245)^2
\]
\[
A = (2.7803)^2 - (0.496)^2
\]
\[
A = 7.728 - 0.246
\]
\[
A \approx 7.482
\]

Therefore, the value of \( A \) is approximately \( 7.482 \).