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The given integral is:

\[
\int \frac{\tan^4(\sqrt{z})}{\sqrt{z} \cdot \cos^2(\sqrt{z})} \, dz
\]

To solve this, let's proceed with a substitution to simplify the integral.

### Step 1: Substitution
Let:
\[
u = \sqrt{z} \quad \text{so} \quad z = u^2
\]
Then:
\[
dz = 2u \, du
\]
Substituting into the integral:
\[
\int \frac{\tan^4(u)}{\sqrt{z} \cdot \cos^2(u)} \cdot 2u \, du
\]
Since \(\sqrt{z} = u\), the integral becomes:
\[
\int \frac{2u \cdot \tan^4(u)}{u \cdot \cos^2(u)} \, du
\]
The \(u\) terms cancel out, so the integral simplifies to:
\[
2 \int \frac{\tan^4(u)}{\cos^2(u)} \, du
\]

### Step 2: Express in terms of trigonometric identities
We know:
\[
\frac{1}{\cos^2(u)} = \sec^2(u)
\]
Thus, the integral becomes:
\[
2 \int \tan^4(u) \cdot \sec^2(u) \, du
\]

### Step 3: Break down \(\tan^4(u)\)
We can express \(\tan^4(u)\) as:
\[
\tan^4(u) = (\tan^2(u))^2
\]
Now, recall that \(\tan^2(u) = \sec^2(u) - 1\), so:
\[
\tan^4(u) = (\sec^2(u) - 1)^2
\]
Expanding this gives:
\[
\tan^4(u) = \sec^4(u) - 2\sec^2(u) + 1
\]

### Step 4: Substitute back and split the integral
Now, the integral becomes:
\[
2 \int (\sec^4(u) - 2\sec^2(u) + 1) \cdot \sec^2(u) \, du
\]
This can be split into three separate integrals:
\[
2 \left( \int \sec^6(u) \, du - 2 \int \sec^4(u) \, du + \int \sec^2(u) \, du \right)
\]

Each of these integrals is a known trigonometric integral. Let's evaluate them:

- \(\int \sec^2(u) \, du = \tan(u)\)
- \(\int \sec^4(u) \, du = \frac{1}{3} \tan(u) + \frac{2}{3} \tan(u) \sec^2(u)\) (using reduction formulas)
- \(\int \sec^6(u) \, du\) is more complicated, but can also be evaluated using reduction formulas or by expressing it in terms of lower powers.

However, the exact expression will lead to a complex combination of trigonometric functions.

### Final Answer
Without further reduction for the higher-power secant integrals, the result will involve a combination of \(\tan(u)\) and \(\sec(u)\) functions with coefficients based on the reduction formulas.

Would you like to go into the detailed solution for \(\int \sec^6(u) \, du\) and \(\int \sec^4(u) \, du\)?

Learn how to solve the integral \(\int \frac{\tan^4(\sqrt{z})}{\sqrt{z} \cdot \cos^2(\sqrt{z})} \, dz\) step-by-step with a detailed solution. This problem involves integration using substitution and trigonometric identities, suitable for advanced mathematics.

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